TANGENT & NORMALTHINGS TO REMEMBER :I The value of the derivative at P (x1 , y1) gives the

slope of the tangent to the curve at P. Symbolically

f (x1) =11yxxd

yd

= Slope of tangent at

P (x1 y1) = m (say).II Equation of tangent at (x1, y1) is ;

y y1 =dydx x y

1 1

(x x1).

III Equation of normal at (x1, y1) is ;

y y1 = 1

1 1

dydx x y

(x x1).

NOTE :1. The point P (x1 ,y1) will satisfy the equation of the curve & the equation of tangent & normal line.2. If the tangent at any point P on the curve is parallel to the axis of x then dy/dx = 0 at the point P.3. If the tangent at any point on the curve is parallel to the axis of y, then dy/dx = or dx/dy = 0.4. If the tangent at any point on the curve is equally inclined to both the axes then dy/dx = ± 1.5. If the tangent at any point makes equal intercept on the coordinate axes then dy/dx = – 1.6. Tangent to a curve at the point P (x1, y1) can be drawn even through dy/dx at P does not exist.

e.g. x = 0 is a tangent to y = x2/3 at (0, 0).7. If a curve passing through the origin be given by a rational integral algebraic equation, the equation of the

tangent (or tangents) at the origin is obtained by equating to zero the terms of the lowest degree in the equation.e.g. If the equation of a curve be x2 – y2 + x3 + 3 x2 y y3 = 0, the tangents at the origin are given byx2 – y2 = 0 i.e. x + y = 0 and x y = 0.

IV Angle of intersection between two curves is defined as the angle between the 2 tangents drawn to the2 curves at their point of intersection. If the angle between two curves is 90° every where then they arecalled ORTHOGONAL curves.

V (a) Length of the tangent (PT) =

)x(f)x(f1y

1

211

(b) Length of Subtangent (MT) =

)x(fy

1

1

(c) Length of Normal (PN) = 211 )x(f1y (d) Length of Subnormal (MN) = y1 f ' (x1)

VI DIFFERENTIALS :The differential of a function is equal to its derivative multiplied by the differential of the independentvariable. Thus if, y = tan x then dy = sec2 x dx.In general dy = f (x) d x.Note that : d (c) = 0 where 'c' is a constant.d (u + v w) = du + dv dw d (u v) = u d v + v d u

Note :1. For the independent variable 'x' , incrementx and differential dx are equal but this is not the case with

the dependent variable 'y' i.e. y d y.

2. The relation dy = f (x) dx can be written asdxdy

= f (x) ; thus the quotient of the differentials of 'y' and

'x' is equal to the derivative of 'y' w.r.t. 'x'.

EXERCISE–IQ.1 Find the equations of the tangents drawn to the curve y2 – 2x3 – 4y + 8 = 0 from the point (1, 2).

Q.2 Find the point of intersection of the tangents drawn to the curve x2y = 1 – y at the points where it isintersected by the curve xy = 1 – y.

Q.3 Find all the lines that pass through the point (1, 1) and are tangent to the curve represented parametricallyas x = 2t – t2 and y = t + t2.

Q.4 In the curve xa yb = Ka+b , prove that the portion of the tangent intercepted between the coordinate axesis divided at its point of contact into segments which are in a constant ratio. (All the constants beingpositive).

Q.5 A straight line is drawn through the origin and parallel to the tangent to a curve

ayax 22 = ln

y

yaa 22

at an arbitary point M. Show that the locus of the point P of

intersection of the straight line through the origin & the straight line parallel to the x-axis & passingthrough the point M is x2 + y2 = a2.

Q.6 Prove that the segment of the tangent to the curve y = 2a

ln 22

22

xaa

xaa

– 22 xa contained between

the y-axis & the point of tangency has a constant length.

Q.7 A function is defined parametrically by the equations

f(t) = x =

2 1 0

0 0

2t ttif t

if t

sin and g(t) = y =

1 0

0

2

tt if t

o if t

sin

Find the equation of the tangent and normal at the point for t = 0 if exist.Q.8 Find all the tangents to the curve y = cos (x + y), 2 x 2, that are parallel to the line x + 2y = 0.

Q.9 (a) Find the value of n so that the subnormal at any point on the curve xyn = an + 1 may be constant.(b) Show that in the curve y = a. ln (x²a²), sum of the length of tangent & subtangent varies as the

product of the coordinates of the point of contact.

Q.10 Prove that the segment of the normal to the curve x = 2a sin t + a sin t cos2t ; y = a cos3t containedbetween the co-ordinate axes is equal to 2a.

Q.11 Show that the normals to the curve x = a (cos t + t sin t) ; y = a (sin t t cos t) are tangent lines to thecircle x2 + y2 = a2.

Q.12 The chord of the parabola y = a2x2 + 5ax 4 touches the curve y =x1

1

at the point x = 2 and is

bisected by that point. Find 'a'.

Q.13 If the tangent at the point (x1, y1) to the curve x3 + y3 = a3 (a 0) meets the curve again in (x2, y2) then

show that1

2

1

2yy

xx

= 1.

Q.14 Determine a differentiable function y = f (x) which satisfies f ' (x) = [f(x)]2 and f (0) = – 21

. Find also the

equation of the tangent at the point where the curve crosses the y-axis.

Q.15 If p1 & p2 be the lengths of the perpendiculars from the origin on the tangent & normal respectively at

any point (x, y) on a curve, then show that

sinycosxpcosysinxp

2

1 where tan = dxdy

. If in the

above case, the curve be x2/3 + y2/3 = a2/3 then show that : 4 p12 + p2

2 = a2.

Q.16 The curve y = ax3 + bx2 + cx + 5 , touches the x -axis at P (2 , 0) & cuts the y-axis at a point Q whereits gradient is 3. Find a , b , c.

Q.17 The tangent at a variable point P of the curve y = x2 x3 meets it again at Q. Show that the locus of themiddle point of PQ is y = 1 9x + 28x2 28x3.

Q.18 Show that the distance from the origin of the normal at any point of the curve

x = a e

2cos2

2sin & y = ae

2sin2

2cos is twice the distance of the tangent at the point

from the origin.Q.19 Show that the condition that the curves x2/3 + y2/3 = c2/3 & (x2/a2) + (y2/b2) = 1 may touch if c = a + b.

Q.20 The graph of a certain function f contains the point (0, 2) and has the property that for each number 'p'the line tangent to y = f (x) at )p(,p f intersect the x-axis at p + 2. Find f (x).

Q.21 A curve is given by the equations x = at2 & y = at3. A variable pair of perpendicular lines through theorigin 'O' meet the curve at P & Q. Show that the locus of the point of intersection of the tangents at P &Q is 4y2 = 3ax a2.

Q.22(a) Show that the curves1

2

2

12

2

Kby

Kax

= 1 &

22

2

22

2

Kby

Kax

= 1 intersect orthogonally..

(b) Find the condition that the curvesby

ax 22

= 1 &by

ax 22

= 1 may cut orthogonally..

Q.23 Show that the angle between the tangent at any point 'A' of the curve ln (x2 + y2) = C tan–1yx and the

line joining A to the origin is independent of the position of A on the curve.

Q.24 For the curve x2/3 + y2/3 = a2/3, show that 2z + 3p2 = a2 where z = x + i y & p is the length of theperpendicular from (0 ,0) to the tangent at (x , y) on the curve.

Q.25 A and B are points of the parabola y = x2. The tangents at A and B meet at C. The median of the triangleABC from C has length 'm' units. Find the area of the triangle in terms of 'm'.

EXERCISE–IIRATE MEASUREANDAPPROXIMATIONS

Q.1 Water is being poured on to a cylindrical vessel at the rate of 1 m3/min. If the vessel has a circular baseof radius 3m, find the rate at which the level of water is rising in the vessel.

Q.2 A man 1.5 m tall walks away from a lamp post 4.5 m high at the rate of 4 km/hr.(i) how fast is the farther end of the shadow moving on the pavement ?(ii) how fast is his shadow lengthening ?

Q.3 A particle moves along the curve 6y = x3 + 2. Find the points on the curve at which the y coordinate ischanging 8 times as fast as the x coordinate.

Q.4 An inverted cone has a depth of 10 cm & a base of radius 5 cm. Water is poured into it at the rate of1.5 cm3/min. Find the rate at which level of water in the cone is rising, when the depth of water is 4cm.

Q.5 A water tank has the shape of a right circular cone with its vertex down. Its altitude is 10 cm and theradius of the base is 15 cm. Water leaks out of the bottom at a constant rate of 1cu. cm/sec. Water ispoured into the tank at a constant rate of C cu. cm/sec. Compute C so that the water level will be risingat the rate of 4 cm/sec at the instant when the water is 2 cm deep.

Q.6 Sand is pouring from a pipe at the rate of 12 cc/sec. The falling sand forms a cone on the ground in sucha way that the height of the cone is always 1/6th of the radius of the base. How fast is the height of thesand cone increasing when the height is 4 cm.

Q.7 An open Can of oil is accidently dropped into a lake ; assume the oil spreads over the surface as acircular disc of uniform thickness whose radius increases steadily at the rate of 10 cm/sec. At the momentwhen the radius is 1 meter, the thickness of the oil slick is decreasing at the rate of 4 mm/sec, how fast isit decreasing when the radius is 2 meters.

Q.8 Water is dripping out from a conical funnel of semi vertical angle/4, at the uniform rate of 2 cm3/secthrough a tiny hole at the vertex at the bottom. When the slant height of the water is 4cm, find the rate ofdecrease of the slant height of the water.

Q.9 An air force plane is ascending vertically at the rate of 100 km/h. If the radius of the earth is R Km, howfast the area of the earth, visible from the plane increasing at 3min after it started ascending. Take visible

area A =hRhR2 2

Where h is the height of the plane in kms above the earth.

Q.10 A variable ABC in the xy plane has its orthocentre at vertex 'B' , a fixed vertex 'A' at the origin and the

third vertex 'C' restricted to lie on the parabola y = 1 +36x7 2

. The point B starts at the point (0, 1) at time

t = 0 and moves upward along the y axis at a constant velocity of 2 cm/sec. How fast is the area of the

triangle increasing when t =27

sec.

Q.11 A circular ink blot grows at the rate of 2 cm2 per second. Find the rate at which the radius is increasing

after 2116 seconds. Use =

722

.

Q.12 Water is flowing out at the rate of 6 m3/min from a reservoir shaped like a hemispherical bowl of radius

R = 13 m. The volume of water in the hemispherical bowl is given by V =  )yR3(y·3

2

when the

water is y meter deep. Find(a) At what rate is the water level changing when the water is 8 m deep.(b) At what rate is the radius of the water surface changing when the water is 8 m deep.

Q.13 If in a triangle ABC, the side 'c' and the angle 'C' remain constant, while the remaining elements are

changed slightly, show thatBcos

dbAcos

da = 0.

Q.14 At time t > 0, the volume of a sphere is increasing at a rate proportional to the reciprocal of its radius. Att = 0, the radius of the sphere is 1 unit and at t = 15 the radius is 2 units.

(a) Find the radius of the sphere as a function of time t.(b) At what time t will the volume of the sphere be 27 times its volume at t = 0.

Q.15 Use differentials to a approximate the values of ; (a) 2.25 and (b) 3 26 .

EXERCISE–III

Q.1 Find the acute angles between the curves y =x2 1 and y =x2 3 at their point of intersection.[ REE '98, 6 ]

Q.2 Find the equation of the straight line which is tangent at one point and normal at another point of thecurve, x = 3t2 , y = 2t3. [ REE 2000 (Mains) 5 out of 100 ]

Q.3 If the normal to the curve, y = f(x) at the point (3, 4) makes an angle 34 with the positive x–axis. Then

f (3) =

(A) – 1 (B) –34 (C) 4

3(D) 1

[JEE 2000 (Scr.) 1 out of 35 ]

Q.4 The point(s) on the curve y3 + 3x2 = 12y where the tangent is vertical, is(are)

(A)

43

2, (B)

1131, (C) (0, 0) (D)

432,

[JEE 2002 (Scr.), 3]Q.5 Tangent to the curve y = x2 + 6 at a point P (1, 7) touches the circle x2 + y2 + 16x + 12y + c = 0 at a point

Q. Then the coordinates of Q are(A) (– 6, –11) (B) (–9, –13) (C) (– 10, – 15) (D) (–6, –7)

[JEE 2005 (Scr.), 3]

MONOTONOCITY(Significance of the sign of the first order derivative)

DEFINITIONS :1. A function f(x) is called an Increasing Function at a point x=a if in a sufficiently small neighbourhood

around x =a we havef a h f a andf a h f a( ) ( )( ) ( )

increasing;

Similarly decreasing iff a h f a andf a h f a( ) ( )( ) ( )

decreasing.

2. A differentiable function is called increasing in an interval (a, b) if it is increasing at every point within theinterval (but not necessarily at the end points). A function decreasing in an interval(a, b) is similarly defined.

3. A function which in a given interval is increasing or decreasing is called“Monotonic” in that interval.4. Tests for increasing and decreasing of a function at a point :

If the derivative f (x) is positive at a point x = a, then the function f (x) at this point is increasing. If it isnegative, then the function is decreasing. Even if f'(a) is not defined, f can still be increasing or decreasing.

Note : If f (a) = 0, then for x = a the function may be still increasing or it may be decreasing as shown. It has tobe identified by a seperate rule. e.g. f (x) = x3 is increasing at every point.Note that, dy/dx = 3 x².

5. Tests for Increasing & Decreasing of a function in an interval :SUFFICIENCY TEST : If the derivative function f (x) in an interval (a ,b) is every where positive, then thefunction f(x) in this interval is Increasing ;If f (x) is every where negative, then f (x) is Decreasing.

General Note :(1) If a function is invertible it has to be either increasing or decreasing.(2) If a function is continuous the intervals in which it rises and falls may be separated by points at which its

derivative fails to exist.(3) If f is increasing in [a, b] and is continuous then f (b) is the greatest and f (c) is the least value of f in [a,

b]. Similarly if f is decreasing in [a, b] then f (a) is the greatest value and f (b) is the least value.6.(a) ROLLE'S THEOREM :

Let f(x) be a function of x subject to the following conditions :(i) f(x) is a continuous function of x in the closed interval of a x b.(ii) f (x) exists for every point in the open interval a < x < b.(iii) f (a) = f (b).

Then there exists at least one point x = c such that a < c < b where f (c) = 0.Note that if f is not continuous in closed [a, b] then it may lead to the adjacentgraph where all the 3 conditions of Rolles will be valid but the assertion will notbe true in (a, b).

disregards whether f isnon derivable or evendiscontinuous at x = a

(b) LMVT THEOREM :Let f(x) be a function of x subject to the following conditions :

(i) f(x) is a continuous function of x in the closed interval of a x b.(ii) f (x) exists for every point in the open interval a < x < b.(iii) f(a) f(b).

Then there exists at least one point x = c such that a < c < b where f (c) = f b f ab a( ) ( )

Geometrically, the slope of the secant line joining the curve at x = a & x = b is equal to the slope of thetangent line drawn to the curve at x = c. Note the following :

Rolles theorem is a special case of LMVT since

f (a) = f (b) f (c) = f b f ab a( ) ( )

= 0.

Note : Now [f (b) – f (a)] is the change in the function f as x changes from a to b so that [f (b) – f (a)] / (b – a)is the average rate of change of the function over the interval [a, b]. Also f '(c) is the actual rate ofchange of the function for x = c. Thus, the theorem states that the average rate of change of a functionover an interval is also the actual rate of change of the function at some point of the interval. In particular,for instance, the average velocity of a particle over an interval of time is equal to the velocity at someinstant belonging to the interval.This interpretation of the theorem justifies the name "Mean Value" for the theorem.

(c) APPLICATION OF ROLLES THEOREM FOR ISOLATING THE REAL ROOTS OF AN EQUATION f (x)=0Suppose a & b are two real numbers such that ;

(i) f(x) & its first derivative f (x) are continuous for a x b.(ii) f(a) & f(b) have opposite signs.(iii) f (x) is different from zero for all values of x between a & b.

Then there is one & only one real root of the equation f(x) = 0 between a & b.

EXERCISE–IQ.1 Find the intervals of monotonocity for the following functions & represent your solution set on the number line.

(a) f(x) = 2. x4x2e (b) f(x) = ex/x (c) f(x) = x2 ex (d) f (x) = 2x2 – ln | x |

Also plot the graphs in each case.

Q.2 Let f (x) = 1 – x – x3. Find all real values of x satisfying the inequality, 1 – f (x) – f 3(x) > f (1 – 5x)

Q.3 Find the intervals of monotonocity of the function(a) f (x) = sin x – cos x in x [0 , 2] (b) g (x) = 2 sinx + cos 2x in (0 x 2).

Q.4 Show that, x3 3x2 9 x + 20 is positive for all values of x > 4.

Q.5 Let f (x) = x3 x2 + x + 1 and g(x) =

2x1,x3

1x0,}xt0:)t(f{max

Discuss the conti. & differentiability of g(x) in the interval (0,2).

Q.6 Find the set of all values of the parameter 'a' for which the function,f(x) = sin 2x – 8(a + 1)sin x + (4a2 + 8a – 14)x increases for all x R and has no critical pointsfor all x R.

Q.7 Find the greatest & the least values of the following functions in the given interval if they exist.

(a) f (x) = sin1

1x

x2

ln x in

3,3

1 (b) y = xx in (0,) (c) y = x5 – 5x4 + 5x3 + 1 in [1, 2]

Q.8 Find the values of 'a' for which the function f(x) = sinx asin2x31

sin3x + 2ax increases throughout the

number line.

Q.9 Prove that f (x) = xe

2

2 dt17)tn2cos(25)tn2(cos9 ll is always an increasing function of x, xR

Q.10 If f(x) =

3

1a2x3 + (a -1)x2 + 2x + 1 is monotonic increasing for every xR then find the range of

values of ‘a’.

Q . 1 1 F i n d t h e s e t o f v a l u e s o f ' a ' f o r w h i c h t h e f u n c t i o n ,

f ( x ) = 121 4

1

2

a aa

x3 + 5x + 7 is increasing at every point of its domain.

Q.12 Find the intervals in which the function f (x) = 3 cos4 x + 10 cos3 x + 6 cos2 x – 3, 0 x ; ismonotonically increasing or decreasing.

Q.13 Find the range of values of 'a' for which the function f (x) = x3 + (2a + 3)x2 + 3(2a + 1)x + 5 is monotonicin R. Hence find the set of values of 'a' for which f (x) in invertible.

Q.14 Find the value of x > 1 for which the function

F (x) =

2x

x

dt32

1tnt1 l is increasing and decreasing.

Q.15 Find all the values of the parameter 'a' for which the function ;f(x) = 8ax a sin 6x 7x sin5x increases & has no critical points for all x R.

Q.16 If f (x) = 2ex – ae–x + (2a+1)x 3 monotonically increases for every x R then find the range of valuesof ‘a’.

Q.17 Construct the graph of the function f (x) =1x

2x3x9x2

and comment upon the following

(a) Range of the function,(b) Intervals of monotonocity,(c) Point(s) where f is continuous but not diffrentiable,(d) Point(s) where f fails to be continuous and nature of discontinuity.(e) Gradient of the curve where f crosses the axis of y.

Q.18 Prove that, x2 – 1 > 2x ln x > 4(x – 1) – 2 ln x for x > 1.

Q.19 Prove that tan2x + 6 ln secx + 2cos x + 4 > 6 sec x for x 322,

.

Q.20 If ax² + (b/x) c for all positive x where a > 0 & b > 0 then show that 27ab2 4c3.

Q.21 If 0 < x < 1 prove that y = x ln x – (x²/2) + (1/2) is a function such that d2y/dx2 > 0. Deducethat x ln x > (x2/2) (1/2).

Q.22 Prove that 0 < x. sin x (1/2) sin² x < (1/2) ( 1) for 0 < x < /2.

Q.23 Show that x² > (1 + x) [ln(1 + x)]2 x > 0.

Q.24 Find the set of values of x for which the inequality ln(1+x) > x/(1+x) is valid.

Q.25 If b > a, find the minimum value of (xa)3+(x b)3, x R.

EXERCISE–IIQ.1 Verify Rolles throrem for f(x) = (x a)m (x b)n on [a, b] ; m, n being positive integer.

Q.2 Let f : [a, b] R be continuous on [a, b] and differentiable on (a, b). If f (a) < f (b), then show thatf ' (c) > 0 for some c (a, b).

Q.3 Let f (x) = 4x3 3x2 2x + 1, use Rolle's theorem to prove that there exist c, 0< c <1 such that f(c) = 0.

Q.4 Using LMVT prove that : (a) tan x > x in

2,0 , (b) sin x < x for x > 0

Q.5 Prove that if f is differentiable on [a, b] and if f (a) = f (b) = 0 then for any real there is an x  (a, b)such that f (x) + f ' (x) = 0.

Q.6 For what value of a, m and b does the function f (x) =

2x1bmx1x0ax3x

0x32

satisfy the hypothesis of the mean value theorem for the interval [0, 2].

Q.7 Suppose that on the interval [–2, 4] the function f is differentiable, f (–2) = 1 and | f ' (x) | 5. Find thebounding functions of f on [–2, 4], using LMVT.

Q.8 Let f, g be differentiable on R and suppose that f (0) = g (0) and f ' (x) g ' (x) for all x 0. Show thatf (x) g (x) for all x 0.

Q.9 Let f be continuous on [a, b] and differentiable on (a, b). If f (a) = a and f (b) = b, show that there existdistinct c1, c2 in (a, b) such that f ' (c1) + f '(c2) = 2.

Q.10 Let f (x) and g (x) be differentiable functions such that f ' (x) g (x) f (x) g ' (x) for any real x. Show thatbetween any two real solutions of f (x) = 0, there is at least one real solution of g (x) = 0.

Q.11 Let f defined on [0, 1] be a twice differentiable function such that, | f " (x) | 1 for all x [0, 1]If f (0) = f (1), then show that, | f ' (x) | < 1 for all x [0, 1]

Q.12 f (x) and g (x) are differentiable functions for 0 x 2 such that f (0) = 5, g (0) = 0, f (2) = 8, g (2) = 1.Show that there exists a number c satisfying 0 < c < 2 and f ' (c) = 3 g' (c).

Q.13 If f,, are continuous in [a, b] and derivable in ]a, b[ then show that there is a value of c lying betweena & b such that,

)c()b()a()c()b()a()c(f)b(f)a(f

= 0

Q.14 Show that exactly two real values of x satisfy the equation x2 = x sinx + cos x.

Q . 1 5 Let a > 0 and f be continuous in [–a, a]. Suppose that f ' (x) exists and f ' (x) 1 for all x (–a, a). Iff (a) = a and f (– a) = – a, show that f (0) = 0.

Q.16 Let a, b, c be three real number such that a < b < c, f (x) is continuous in [a, c] and differentiablein (a, c). Also f ' (x) is strictly increasing in (a, c). Prove that

(c – b) f (a) + (b – a) f (c) > (c – a) f (b)

Q.17 Use the mean value theorem to prove,x

1x < ln x < x – 1, x > 1

Q.18 Use mean value theorem to evaluate, x1xLimx

.

Q.19 Using L.M.V.T. or otherwise prove that difference of square root of two consecutive natural numbers

greater than N2 is less thanN21

.

Q.20 Prove the inequality ex > (1 + x) using LMVT for all x R0 and use it to determine which of the twonumbers e and e is greater.

EXERCISE–IIIQ.1 If f (x) =

xsin

x & g (x) =

xtan

x , where 0 < x 1, then in this interval :

(A) both f (x) & g(x) are increasing functions (B) both f (x) & g(x) are decreasing functions(C) f (x) is an increasing function (D) g(x) is an increasing function

[ JEE '97 (Scr), 2 ]

Q.2 Let a + b = 4 , where a < 2 and let g (x) be a differentiable function . Ifdgdx

> 0 for all x, prove that

g x dx g x dxba

( ) ( ) 00

increases as (b a) increases. [JEE ’97, 5]

Q.3(a) Let h(x) = f(x) (f(x))2 + (f(x))3 for every real number x. Then :(A) h is increasing whenever f is increasing (B) h is increasing whenever f is decreasing(C) h is decreasing whenever f is decreasing (D) nothing can be said in general.

(b) f(x) = xx

2

211

, for every real number x, then the minimum value of f :

(A) does not exist because f is unbounded (B) is not attained even though f is bounded(C) is equal to 1 (D) is equal to 1. [ JEE '98, 2 + 2 ]

Q.4(a) For all x (0, 1) :(A) ex < 1 + x (B) loge(1 + x) < x (C) sin x > x (D) loge x > x

(b) Consider the following statements S and R :S : Both sin x & cos x are decreasing functions in the interval (/2,).R : If a differentiable function decreases in an interval (a, b), then its derivative also decreases in (a, b).Which of the following is true ?(A) both S and R are wrong(B) both S and R are correct, but R is not the correct explanation for S(C) S is correct and R is the correct explanation for S(D) S is correct and R is wrong.

(c) Let f (x) = ex (x 1) (x 2) d x then f decreases in the interval :(A) (, 2) (B) ( 2, 1) (C) (1, 2) (D) (2, +)

[JEE 2000 (Scr.) 1+1+1 out of 35]

Q.5(a) If f (x) = xex(1 – x), then f(x) is

(A) increasing on

1,

21

(B) decreasing on R

(C) increasing on R (D) decreasing on

121,

(b) Let – 1 < p < 1. Show that the equation 4x3 – 3x – p = 0 has a unique root in the interval12, 1

and

identify it. [ JEE 2001, 1 + 5 ]

Q.6 The length of a longest interval in which the function 3sinx – 4sin3x is increasing, is

(A)3

(B)2

(C)32

(D)

[JEE 2002 (Screening), 3]

Q.7(a) Using the relation 2(1 – cosx) < x2 , x 0 or otherwise, prove that sin (tanx) > x , x

4,0 .

(b) Let f : [0, 4] R be a differentiable function.(i) Show that there exist a, b [0, 4], (f (4))2 – (f (0))2 = 8 f (a) f (b)(ii) Show that there exist , with 0 << < 2 such that

4

0f(t) dt = 2 ( f (2) + f (2) ) [JEE 2003 (Mains), 4 + 4 out of 60]

Q.8(a) Let f (x) =

0x,0

0x,nxx l. Rolle’s theorem is applicable to f for x[0, 1], if=

(A) –2 (B) –1 (C) 0 (D) 21

(b) If f is a strictly increasing function, then )0(f)x(f)x(f)x(fLim

2

0x

is equal to

(A) 0 (B) 1 (C) –1 (D) 2[JEE 2004 (Scr)]

Q.9 If p (x) = 51x101 – 2323x100 – 45x + 1035, using Rolle's theorem, prove that at least one root of p(x)lies between (451/100, 46). [JEE 2004, 2 out of 60]

Q.10(a) If f (x) is a twice differentiable function and given that f(1) = 1, f(2) = 4, f(3) = 9, then(A) f '' (x) = 2, for x (1, 3) (B) f '' (x) = f ' (x) = 2, for some x (2, 3)(C) f '' (x) = 3, for x (2, 3) (D) f '' (x) = 2, for some x (1, 3)

[JEE 2005 (Scr), 3] (b) f (x) is differentiable function and g (x) is a double differentiable function such that | f (x) | 1 and

f '(x) = g (x). If f2(0) +g2(0) = 9. Prove that there exists some c (–3, 3) such that g (c) · g"(c)<0.[JEE 2005 (Mains), 6]

MAXIMA - MINIMAFUNCTIONS OF A SINGLE VARIABLE

HOW MAXIMA & MINIMAARE CLASSIFIED1. A function f(x) is said to have a maximum

at x = a if f(a) is greater than every othervalue assumed by f(x) in the immediateneighbourhood of x = a. Symbolicallyf a f a hf a f a h( ) ( )( ) ( )

x = a gives maxima for

a sufficiently small positive h.Similarly, a function f(x) is said to have aminimum value at x = b if f(b) is least than everyother value assumed by f(x) in the immediateneighbourhood at x = b. Symbolically iff b f b hf b f b h( ) ( )( ) ( )

x = b gives minima for a sufficiently small positive h.

Note that :(i) the maximum & minimum values of a function are also known as local/relative maxima or

local/relative minima as these are the greatest & least values of the function relative to some neighbourhoodof the point in question.

(ii) the term 'extremum' or (extremal) or 'turning value' is used both for maximum or a minimum value.(iii) a maximum (minimum) value of a function may not be the greatest (least) value in a finite interval.(iv) a function can have several maximum & minimum values & a minimum value may even be greater than a

maximum value.(v) maximum & minimum values of a continuous function occur alternately & between two consecutive

maximum values there is a minimum value & vice versa.2. A NECESSARY CONDITION FOR MAXIMUM & MINIMUM :

If f(x) is a maximum or minimum at x = c & if f (c) exists then f (c) = 0.Note :

(i) The set of values of x for which f (x) = 0 are often called as stationary points or critical points. The rateof change of function is zero at a stationary point.

(ii) In case f (c) does not exist f(c) may be a maximum or a minimum & in this case left hand and right handderivatives are of opposite signs.

(iii) The greatest (global maxima) and the least (global minima) values of a function f in an interval [a, b] aref(a) or f(b) or are given by the values of x for which f (x) = 0.

(iv) Critical points are those where dydx

= 0, if it exists , or it fails to exist either by virtue of a vertical tangentor by virtue of a geometrical sharp corner but not because of discontinuity of function.

3. SUFFICIENT CONDITION FOR EXTREME VALUES :

f (c-h) > 0f (c+ h) < 0

x = c is a point of local maxima, where f (c) = 0.

Similarly

f (c-h) < 0f (c+h) > 0

x = c is a point of local minima, where f (c) = 0.

quantitypositivesmall

lysufficientaish

Note : If f (x) does not change sign i.e. has the same sign in a certain complete neighbourhood of c,then f(x) is either strictly increasing or decreasing throughout this neighbourhood implying that f(c) is notan extreme value of f.

4. USE OF SECOND ORDER DERIVATIVE IN ASCERTAINING THE MAXIMA OR MINIMA:(a) f(c) is a minimum value of the function f, if f (c) = 0 & f (c) > 0.(b) f(c) is a maximum value of the function f, f (c) = 0 & f (c) < 0.

Note : if f (c) = 0 then the test fails. Revert back to the first order derivative check for ascertaning themaxima or minima.

5. SUMMARYWORKING RULE :FIRST :When possible , draw a figure to illustrate the problem & label those parts that are important in theproblem. Constants & variables should be clearly distinguished.SECOND :Write an equation for the quantity that is to be maximised or minimised. If this quantity is denoted by ‘y’,it must be expressed in terms of a single independent variable x. his may require some algebraicmanipulations.THIRD :If y = f (x) is a quantity to be maximum or minimum, find those values of x for whichdy/dx = f (x) = 0.FOURTH :Test each values of x for which f (x) = 0 to determine whether it provides a maximum or minimum orneither. The usual tests are :(a) If d²y/dx² is positive when dy/dx = 0 y is minimum.

If d²y/dx² is negative when dy/dx = 0 y is maximum.If d²y/dx² = 0 when dy/dx = 0, the test fails.

(b) If dydx

ispositive for x xzero for x xnegative for x x

0

0

0

a maximum occurs at x = x0.

But if dy/dx changes sign from negative to zero to positive as x advances throughxo there is a minimum. If dy/dx does not change sign, neither a maximum nor a minimum. Such points arecalled INFLECTION POINTS.FIFTH :If the function y = f (x) is defined for only a limited range of values a x b then examine x = a &x = b for possible extreme values.SIXTH :If the derivative fails to exist at some point, examine this point as possible maximum or minimum.Important Note :– Given a fixed point A(x1, y1) and a moving point P(x, f (x)) on the curve y = f(x). Then AP will be

maximum or minimum if it is normal to the curve at P.– If the sum of two positive numbers x and y is constant than their product is maximum if they are

equal, i.e. x + y = c , x > 0 , y > 0 , then

xy = 41

[ (x + y)2 – (x – y)2 ]

– If the product of two positive numbers is constant then their sum is least if they are equal.i.e. (x + y)2 = (x – y)2 + 4xy

6. USEFUL FORMULAE OF MENSURATION TO REMEMBER : Volume of a cuboid = lbh. Surface area of a cuboid = 2 (lb + bh + hl). Volume of a prism = area of the base x height. Lateral surface of a prism = perimeter of the base x height. Total surface of a prism = lateral surface + 2 area of the base

(Note that lateral surfaces of a prism are all rectangles).

Volume of a pyramid =13

area of the base x height.

Curved surface of a pyramid =12

(perimeter of the base) x slant height.

(Note that slant surfaces of a pyramid are triangles).

Volume of a cone =13 r2h.

Curved surface of a cylinder = 2 rh.

Total surface of a cylinder = 2 rh + 2 r2.

Volume of a sphere =43 r3.

Surface area of a sphere = 4 r2.

Area of a circular sector =12

r2 , when is in radians.

7. SIGNIFICANCE OF THE SIGN OF 2ND ORDER DERIVATIVE AND POINTS OF INFLECTION :The sign of the 2nd order derivative determines the concavity ofthe curve. Such points such as C & E on the graph where theconcavity of the curve changes are called the points of inflection.From the graph we find that if:

(i)d ydx

2

2 > 0 concave upwards

(ii)d ydx

2

2 < 0 concave downwards.

At the point of inflection we find thatd ydx

2

2 = 0 &

d ydx

2

2 changes sign.

Inflection points can also occur ifd ydx

2

2 fails to exist. For example, consider the graph of the function

defined as,

f (x) = [ x for xx for x

3 5

2

12 1

/ ( , )( , )

Note that the graph exhibits two critical points one is a point of localmaximum & the other a point of inflection.

EXERCISE–I

Q.1 A cubic f(x) vanishes at x = 2 & has relative minimum/maximum at x = 1 and x = 31

.

If

1

1

dx)x(f = 314

, find the cubic f (x).

Q.2 Investigate for maxima & minima for the function, f (x) =1

x

[2 (t 1) (t 2)3 + 3 (t 1)2 (t 2)2] dt

Q.3 Find the maximum & minimum value for the function ;(a) y = x + sin 2x , 0 x 2 (b) y = 2 cos 2x cos 4x , 0 x

Q.4 Suppose f(x) is real valued polynomial function of degree 6 satisfying the following conditions ;(a) f has minimum value at x = 0 and 2(b) f has maximum value at x = 1

(c) for all x, Limitx 0

1x ln

x1

x1

x)x(f

011001

= 2.

Determine f (x).

Q.5 Find the maximum perimeter of a triangle on a given base ‘a’ and having the given vertical angle.

Q.6 The length of three sides of a trapezium are equal, each being 10 cms. Find the maximum area of such atrapezium.

Q .7 The plan view of a swimming pool consists of a semicircle of radius r attached to a rectangle of length '2r'and width 's'. If the surface area A of the pool is fixed, for what value of 'r' and 's' the perimeter 'P' of thepool is minimum.

Q.8 For a given curved surface of a right circular cone when the volume is maximum, prove that the semi

vertical angle is sin1 13

.

Q.9 Of all the lines tangent to the graph of the curve y =3x

62

, find the equations of the tangent lines of

minimum and maximum slope.

Q.10 A statue 4 metres high sits on a column 5.6 metres high. How far from the column must a man, whose eyelevel is 1.6 metres from the ground, stand in order to have the most favourable view of statue.

Q.11 By the post office regulations, the combined length & girth of a parcel must not exceed 3 metre. Find thevolume of the biggest cylindrical (right circular) packet that can be sent by the parcel post.

Q.12 A running track of 440 ft. is to be laid out enclosing a football field, the shape of which is a rectangle withsemi circle at each end. If the area of the rectangular portion is to be maximum, find the length of its sides.

Use : 722

.

Q.13 A window of fixed perimeter (including the base of the arch) is in the form of a rectangle surmounted bya semicircle. The semicircular portion is fitted with coloured glass while the rectangular part is fitted withclean glass. The clear glass transmits three times as much light per square meter as the coloured glassdoes. What is the ratio of the sides of the rectangle so that the window transmits the maximum light?

Q.14 A closed rectangular box with a square base is to be made to contain 1000 cubic feet. The cost of thematerial per square foot for the bottom is 15 paise, for the top 25 paise and for the sides 20 paise. Thelabour charges for making the box are Rs. 3/-. Find the dimensions of the box when the cost is minimum.

Q.15 Find the area of the largest rectangle with lower base on the x-axis & upper vertices on thecurve y = 12 x2.

Q.16 A trapezium ABCD is inscribed into a semicircle of radius l so that the base AD of the trapezium is adiameter and the vertices B & C lie on the circumference. Find the base angle of the trapezium ABCDwhich has the greatest perimeter.

Q.17 If y = ax bx x

( ) ( )1 4

has a turning value at (2, 1) find a & b and show that the turning value is a

maximum.

Q.18 Prove that among all triangles with a given perimeter, the equilateral triangle has the maximum area.

Q.19 A sheet of poster has its area 18 m². The margin at the top & bottom are 75 cms and at the sides50 cms. What are the dimensions of the poster if the area of the printed space is maximum?

Q.20 A perpendicular is drawn from the centre to a tangent to an ellipsexa

2

2 +yb

2

2 = 1. Find the greatest value

of the intercept between the point of contact and the foot of the perpendicular.

Q.21 Consider the function, F (x) =

x

1

2 dt)tt( , x R.

(a) Find the x and y intercept of F if they exist.(b) Derivatives F ' (x) and F '' (x).(c) The intervals on which F is an increasing and the invervals on which F is decreasing.(d) Relative maximum and minimum points.(e) Any inflection point.

Q.22 A beam of rectangular cross section must be sawn from a round log of diameter d. What should thewidth x and height y of the cross section be for the beam to offer the greatest resistance (a) to compression;(b) to bending. Assume that the compressive strength of a beam is proportional to the area of the crosssection and the bending strength is proportional to the product of the width of section by the square of itsheight.

Q.23 What are the dimensions of the rectangular plot of the greatest area which can be laid out within a triangleof base 36 ft. & altitude 12 ft ? Assume that one side of the rectangle lies on the base of the triangle.

Q.24 The flower bed is to be in the shape of a circular sector of radius r & central angle . If the area is fixed& perimeter is minimum, find r and.

Q.25 The circle x2 + y2 = 1 cuts the x-axis at P & Q. Another circle with centre at Q and varable radiusintersects the first circle at R above the x-axis & the line segment PQ at S. Find the maximum area ofthe triangle QSR.

EXERCISE–II

Q.1 The mass of a cell culture at time t is given by, M (t) = te413

(a) Find )t(MLimt

and )t(MLimt

(b) Show that dtdM

= )M3(M31

(c) Find the maximum rate of growth of M and also the vlaue of t at which occurs.Q.2 Find the cosine of the angle at the vertex of an isosceles triangle having the greatest area for the given

constant length l of the median drawn to its lateral side.

Q.3 From a fixed point A on the circumference of a circle of radius 'a', let the perpendicular AY fall on thetangent at a point P on the circle, prove that the greatest area which the APY can have

is 3 38a2

sq. units.

Q.4 Given two points A (2 ,0) & B (0 ,4) and a line y = x. Find the co-ordinates of a point M on this lineso that the perimeter of the AMB is least.

Q.5 A given quantity of metal is to be casted into a half cylinder i.e. with a rectangular base and semicircularends. Show that in order that total surface area may be minimum , the ratio of the height of the cylinderto the diameter of the semi circular ends is /(+2).

Q.6 Depending on the values of pR, find the value of 'a' for which the equation x3 + 2 px2 + p = a has threedistinct real roots.

Q.7 Show that for each a > 0 the function eax. xa² has a maximum value say F (a), and that F (x) has aminimum value, ee/2.

Q.8 For a > 0, find the minimum value of the integral a1

0

ax253 dxe)xax4a( .

Q.9 Find the maximum value of the integral

1

1

x dxe|ax| where | a | 1.

Q.10 Consider the function f (x) =

0xfor0

0xwhenxnx

l

(a) Find whether f is continuous at x = 0 or not.(b) Find the minima and maxima if they exist.

(c) Does f ' (0) ? Find )x('Lim0x

f

.

(d) Find the inflection points of the graph of y = f (x)..

Q.11 Consider the function y = f (x) = ln (1 + sin x) with – 2 x 2. Find(a) the zeroes of f (x)(b) inflection points if any on the graph(c) local maxima and minima of f (x)(d) asymptotes of the graph(e) sketch the graph of f (x) and compute the value of the definite integral

2

2

dx)x(f .

Q.12 A right circular cone is to be circumscribed about a sphere of a given radius. Find the ratio of the altitudeof the cone to the radius of the sphere, if the cone is of least possible volume.

Q.13 Find the point on the curve 4 x² + a²y² = 4 a², 4 < a² < 8 that is farthest from the point (0 ,2).

Q.14 Find the set of value of m for the cubic x3 – 23

x2 + 25

= )(log 41 m has 3 distinct solutions.

Q.15 Let A(p2,p) , B(q2,q) , C(r2, r) be the vertices of the triangle ABC. A parallelogram AFDE is drawnwith vertices D, E & F on the line segments BC, CA & AB respectively. Using calculus, show that

maximum area of such a parallelogram is : 41

(p + q) (q + r) (p – r).

Q.16 A cylinder is obtained by revolving a rectangle about the xaxis , the base of the rectangle lying on thexaxis and the entire rectangle lying in the region between the curve

y = 1xx2 & the xaxis. Find the maximum possible volume of the cylinder..

Q.17 For what values of ‘a’ does the function f (x) = x3 + 3 (a 7) x2 + 3 (a2 9) x1 have a positive pointof maximum.

Q.18 Among all regular triangular prism with volume V, find the prism with the least sum of lengths of all edges.How long is the side of the base of that prism?

Q.19 A segment of a line with its extremities on AB and AC bisects a triangle ABC with sides a, b, c into twoequal areas. Find the length of the shortest segment.

Q.20 What is the radius of the smallest circular disk large enough to cover every acute isosceles triangle of agiven perimeter L?

Q.21 Find the magnitude of the vertex angle ‘’ of an isosceles triangle of the given area ‘A’ such that theradius ‘r’ of the circle inscribed into the triangle is the maximum.

Q.22 Prove that the least perimeter of an isosceles triangle in which a circle of radius r can be inscribed

is 6 r 3 .

Q.23 The function f (x) defined for all real numbers x has the following properties(i) f (0) = 0, f (2) = 2 and f ' (x) = k(2x – x2)e–x for some constant k > 0. Find(a) the intervals on which f is increasing and decreasing and any local maximum or minimum values.(b) the intervals on which the graph f is concave down and concave up.(c) the function f (x) and plot its graph.

Q.24 Find the minimum value of | sin x + cos x + tan x + cot x + sec x + cosec x | for all real x.

Q.25 Use calculus to prove the inequality, sin x x2

in 0 x 2

.

You may use the inequality to prove that, cos x 1 –

2x in 0 x 2

EXERCISE–IIIQ.1 A conical vessel is to be prepared out of a circular sheet of gold of unit radius. How much sectorial area

is to be removed from the sheet so that the vessel has maximum volume. [ REE '97, 6 ]

Q.2(a) The number of values of x where the function f(x) = cos x + cos 2 x attains its maximum is :

(A) 0 (B) 1 (C) 2 (D) infinite (b) Suppose f(x) is a function satisfying the following conditions :

(i) f(0) = 2, f(1) = 1 (ii) f has a minimum value at x =52 and

(iii) for all x f (x) =bax21b2ax2)bax(2

11bb1bax21ax2ax2

Where a, b are some constants. Determine the constants a, b & the function f(x).[JEE '98, 2 + 8]

Q.3 Find the points on the curve ax2 + 2bxy + ay2 = c ; c > b > a > 0, whose distance from the origin isminimum. [ REE '98, 6]

Q.4 The function f(x) =1

x

t (et 1) (t 1) (t 2)3 (t 3)5 dt has a local minimum at x =

(A) 0 (B) 1 (C) 2 (D) 3[ JEE '99 (Screening), 3]

Q.5 Find the co-ordinates of all the points P on the ellipse (x2/a2) + (y2/b2) = 1 for which the area of thetriangle PON is maximum, where O denotes the origin and N the foot of the perpendicular from O to thetangent at P. [JEE '99, 10 out of 200]

Q.6 Find the normals to the ellipse (x2/9) + (y2/4) = 1 which are farthest from its centre. [REE '99, 6]

Q.7 Find the point on the straight line, y = 2x + 11 which is nearest to the circle,16 (x2 + y2) + 32 x 8 y 50 = 0. [REE 2000 Mains, 3 out of 100]

Q.8 Let f (x) = [ | | | |x for xfor x

0 21 0

. Then at x = 0, ' f ' has :

(A) a local maximum (B) no local maximum(C) a local minimum (D) no extremum.

[ JEE 2000 Screening, 1 out of 35 ]Q.9 Find the area of the right angled triangle of least area that can be drawn so as to circumscribe a rectangle

of sides 'a' and 'b', the right angles of the triangle coinciding with one of the angles of the rectangle.[ REE 2001 Mains, 5 out of 100 ]

Q.10(a) Let f(x) = (1 + b2)x2 + 2bx + 1 and let m(b) be the minimum value of f(x). As b varies, the rangeof m (b) is

(A) [0, 1] (B) 0 12,

(C)121,

(D) (0, 1]

(b) The maximum value of (cos1) · (cos2).......... (cosn), under the restrictions

O < 1, 2,..............., n <2

and cot 1 · cot 2.......... cot n = 1 is

(A)12 2n/ (B)

12n

(C)12n

(D) 1

[ JEE 2001 Screening, 1 + 1 out of 35 ]

Q.11(a) If a1 , a2 ,....... , an are positive real numbers whose product is a fixed number e, the minimum value ofa1 + a2 + a3 +....... + an–1 + 2an is(A) n(2e)1/n (B) (n+1)e1/n (C) 2ne1/n (D) (n+1)(2e)1/n

[ JEE 2002 Screening] (b) A straight line L with negative slope passes through the point (8,2) and cuts the positive coordinates axes

at points P and Q. Find the absolute minimum value of OP + OQ, as L varies, where O is the origin. [ JEE 2002 Mains, 5 out of 60]

Q.12(a) Find a point on the curve x2 + 2y2 = 6 whose distance from the line x + y = 7, is minimum.[JEE-03, Mains-2 out of 60]

(b) For a circle x2 + y2 = r2, find the value of ‘r’ for which the area enclosed by the tangents drawn from thepoint P(6, 8) to the circle and the chord of contact is maximum. [JEE-03, Mains-2 out of 60]

Q.13(a) Let f (x) = x3 + bx2 + cx + d, 0 < b2 < c. Then f(A) is bounded (B) has a local maxima(C) has a local minima (D) is strictly increasing [JEE 2004 (Scr.)]

(b) Prove that

)1x(·x3x2xsin x

2,0 . (Justify the inequality, if any used).

[JEE 2004, 4 out of 60]Q.14 If P(x) be a polynomial of degree 3 satisfying P(–1) = 10, P(1) = – 6 and P(x) has maximum at x = – 1

and P'(x) has minima at x = 1. Find the distance between the local maximum and local minimum of thecurve. [JEE 2005 (Mains), 4 out of 60]

Q.15(a) If f (x) is cubic polynomial which has local maximum at x = – 1. If f (2) = 18, f (1) = – 1 andf '(x) has local maxima at x = 0, then(A) the distance between (–1, 2) and (a, f (a)), where x = a is the point of local minima is 52 .

(B) f (x) is increasing for x [1, 52 ](C) f (x) has local minima at x = 1(D) the value of f(0) = 5

(b) f (x) =

3x2ex2x1e21x0e

1x

x

and g (x) = x

0

dttf , x [1, 3] then g(x) has

(A) local maxima at x = 1 + ln 2 and local minima at x = e(B) local maxima at x = 1 and local minima at x = 2(C) no local maxima(D) no local minima [JEE 2006, 5marks each]

(c) If f (x) is twice differentiable function such that f (a) = 0, f (b) = 2, f (c) = – 1, f (d) = 2, f (e) = 0,

where a < b < c < d < e, then find the minimum number of zeros of )x("f).x(f)x('f)x(g 2 in theinterval [a, e]. [JEE 2006, 6]

ANSWER KEYTANGENT & NORMAL

EXERCISE–I

Q.1 2 3 x y = 2 3 1 or 2 3 x + y = 2 3 1 Q.2 (0, 1)

Q.3 x = 1 when t = 1, m ; 5x – 4y = 1 if t 1, m = 1/3]Q.7 T : x – 2y = 0 ; N : 2x + y = 0 Q.8 x + 2 y = /2 & x + 2 y = 3 /2

Q.9 (a) n = 2 Q.12 a = 1 Q.14 –2x

1

; x – 4y = 2 Q.16 a = 1/2 ; b = 3/4 ; c = 3

Q.20 2e–x/2 Q.22 (b) a b = a b Q.23 = tan–1 2C

Q.252mm

EXERCISE–IIQ.1 1/9 m/min Q.2 (i) 6 km/h (ii) 2 km/hr Q.3 (4 , 11) & ( 4, 31/3)Q.4 3/8 cm/min Q.5 1 + 36 cu. cm/sec Q.6 1/48 cm/s Q.7 0.05 cm/sec

Q.8 24

cm/s Q.9 200r3 / (r + 5)² km² / h Q.10667 Q.11

14 cm/sec.

Q.12 (a) –24

1m/min., (b) –

2885

m/min. Q.14 (a) r = (1 + t)1/4, (b) t = 80 Q.15 (a) 5.02, (b)8027

EXERCISE–III

Q.1 = tan14 27

Q.2 2 x + y 2 2 = 0 or 2 x y 2 2 = 0 Q.3 D Q.4 D Q.5 D* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *

MONOTONOCITYEXERCISE–I

Q.1 (a) I in (2 ,) & D in (, 2) (b) I in (1 ,) & D in ( , 0) (0 , 1)(c) I in (0, 2) & D in (, ) (2 ,)

(d) I for x >12 or 1

2 < x < 0 & D for x < 1

2 or 0 < x < 1

2Q.2 (–2, 0) (2, )Q.3 (a) I in [0, 3/4) (7/4 , 2 ] & D in (3/4 , 7 /4)

(b) I in [0 , /6) (/2 , 5/6) (3/2 , 2 ] & D in (/6 , /2) (5/6, 3 /2)]

Q.5 continuous but not diff. at x = 1 Q.6 a < 2 5 or a > 5

Q.7 (a) (/6)+(1/2)ln 3, (/3) – (1/2)ln3, (b) least value is equal to (1/e)1/e, no greatest value, (c) 2 &10Q.8 [1,) Q.10 a (–, – 3] [1 ,) Q.11 [ 7, 1) [2, 3]Q.12 increasing in x (/2 , 2/3) & decreasing in [0 , /2) (2/3 ,]

Q.13 0 a 23

Q.14 in (3,) and in (1, 3) Q.15 (6,) Q.16 a 0

Q.17 (a) (– , 0] ; (b) in

35,1 and in }3{,

35)1,(

; (c) x = 3

5;

(d) removable discont. at x = –3 (missing point) and non removable discont. at x = 1 (infinite type)(e) – 2

Q.24 ( 1, 0) (,) Q.25 (b a)3/4

EXERCISE–II

Q.1 c = mb nam n

which lies between a & b Q.6 a = 3, b = 4 and m = 1

Q.7 y = – 5x – 9 and y = 5x + 11 Q.18 0

EXERCISE–IIIQ.1 C Q.3 (a) A, C ; (b) D Q.4 (a) B ; (b) D ; (c) C

Q.5 (a) A, (b) cos13

1cos

p Q.6 A Q.8 (a) D ; (b) C Q.10 (a) D

* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *

MAXIMA - MINIMAEXERCISE–I

Q.1 f (x) = x3 + x2 x + 2 Q.2 max. at x = 1 ; f(1) = 0 , min. at x = 7/5 ; f(7/5) = 108/3125Q.3 (a) Max at x = 2, Max value = 2, Min. at x = 0 , Min value = 0

(b) Max at x = /6 & also at x = 5 /6 and Max value = 3/2 , Min at x = /2 , Min value = 3

Q.4 f (x) = 23

x6 125

x5 + 2 x4 Q.5 Pmax = a 12

cosec Q.6 75 3 sq. units

Q.7 r =24A

s =24A

Q.9 3x + 4y – 9 = 0 ; 3x – 4y + 9 = 0

Q.10 4 2 m Q.11 1/ cu mQ.12 110 ' , 70 ' Q.13 6/(6 + ) Q.14 side 10', height 10'Q.15 32 sq. units Q.16 = 600 Q.17 a = 1, b = 0Q.19 width 2 3m, length 3 3 m Q.20 a bQ.21 (a) (–1, 0), (0, 5/6) ; (b) F ' (x) = (x2 – x), F '' (x) = 2x – 1, (c) increasing (– , 0) (1, ),

decreasing (0, 1) ; (d) (0, 5/6) ; (1, 2/3) ; (e) x = 1/2

Q.22 (a) x = y = d2

, (b) x = d3

, y = 23

d Q.23 6' × 18'

Q.24 r = A , = 2 radians Q.25 334

EXERCISE–II

Q.1 (a) 0, 3, (c) 43

, t = ln 4 Q.2 cos A = 0.8 Q.4 (0 , 0)

Q.6 p < a < 3227

3p + p if p > 0 ; 3227

3p + p < a < p if p < 0 Q.8 4 when a = 2

Q.9 Maximum value is (e + e–1) when a = – 1

Q.10 (a) f is continuous at x = 0 ; (b) – e2

; (c) does not exist, does not exist ; (d) pt. of inflection x = 1

Q.11 (a) x = – 2, – , 0, , 2, (b) no inflection point, (c) maxima at x = 2

and – 23

and no minima,

(d) x = 23

and x = – 2

, (e) – ln 2

Q.12 4 Q.13 (0 , 2) & max. distance = 4 Q.14 m

161,

321

Q.16 4

Q.17 (, 3) (3 , 29/7) Q.18 H = x =3/1

3V4

Q.19

2)cba()bac(

Q.20 L/4 Q.213

Q.23 (a) increasing in (0, 2) and decreasing in (–, 0) (2,), local min. value = 0 and local max. value = 2

(b) concave up for (–, 2 – 2 ) (2 + 2 ,) and concave down in (2 – 2 ), (2 + 2 )

(c) f (x) = 2x·2 x·e21

Q.24 122

EXERCISE–III

Q.1

321 sq. units Q.2 (a) B, (b) a = 1

4; b = 5

4 ; f(x) = 1

4(x2 5x + 8)

Q.3ca b

ca b2 2( )

,( )

&

ca b

ca b2 2( )

,( ) Q.4 (a) B, D,

Q.5 ± a

a b

2

2 2, ± b

a b

2

2 2Q.6 ± 3 x ± 2 y = 5 Q.7 (–9/2 , 2)

Q.8 A Q.9 2ab Q.10 (a) D ; (b) A Q.11 (a) A ; (b) 18 Q.12 (a) (2, 1) ; (b) 5Q.13 (a) D Q.14 654 Q.15 (a) B, C; (b) A, B, (c) 6 solutions

* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *

Question bank on Application of DerivativeSelect the correct alternative : (Only one is correct)Q.1 Suppose x1 & x2 are the point of maximum and the point of minimum respectively of the function

f(x) = 2x3 9ax2 + 12a2x + 1 respectively, then for the equality 21x = x2 to be true the value of 'a' must be

(A) 0 (B) 2 (C) 1 (D) 1/4

Q.2 Point 'A' lies on the curve2xey and has the coordinate (x,

2xe ) where x > 0. Point B has thecoordinates (x, 0). If 'O' is the origin then the maximum area of the triangle AOB is

(A) e21

(B) e41

(C) e1

(D) e81

Q.3 The angle at which the curve y = KeKx intersects the y-axis is :(A) tan1 k2 (B) cot1 (k2) (C) sec1 1 4

k

(D) none

Q.4 {a1, a2, ....., a4, ......} is a progression where an =n

n

2

3 200 . The largest term of this progression is :

(A) a6 (B) a7 (C) a8 (D) none

Q.5 The angle between the tangent lines to the graph of the function f (x) = x

2

dt)5t2( at the points where

the graph cuts the x-axis is(A) /6 (B) /4 (C) /3 (D) /2

Q.6 The minimum value of the polynomial x(x + 1) (x + 2) (x + 3) is :(A) 0 (B) 9/16 (C) 1 (D) 3/2

Q.7 The minimum value of tan

tan

x

x

6 is :

(A) 0 (B) 1/2 (C) 1 (D) 3Q.8 The difference between the greatest and the least values of the function, f (x) = sin2x – x on

2,

2

(A) (B) 0 (C)32

3 (D)

32

23

Q.9 The radius of a right circular cylinder increases at the rate of 0.1 cm/min, and the height decreases at therate of 0.2 cm/min. The rate of change of the volume of the cylinder, in cm3/min, when the radius is 2 cmand the height is 3 cm is

(A) – 2 (B) – 58

(C) – 53

(D) 52

Q.10 If a variable tangent to the curve x2y = c3 makes intercepts a, b on x and y axis respectively, then the valueof a2b is

(A) 27 c3 (B) 3c274

(C) 3c4

27(D) 3c

94

Q.11 Difference between the greatest and the least values of the functionf (x) = x(ln x – 2) on [1, e2] is

(A) 2 (B) e (C) e2 (D) 1

Q.12 Let f (x) =

n2

0r

r

n

xtan

xtan , n N, where x

2,0

(A) f (x) is bounded and it takes both of it's bounds and the range of f (x) contains exactly one integral point.(B) f (x) is bounded and it takes both of it's bounds and the range of f (x) contains more than one integral point.(C) f (x) is bounded but minimum and maximum does not exists.(D) f (x) is not bounded as the upper bound does not exist.

Q.13 I f f (x) = x3 + 7x – 1 then f (x) has a zero between x = 0 and x = 1. The theorem which best describes this,is

(A) Squeeze play theorem (B) Mean value theorem(C) Maximum-Minimum value theorem (D) Intermediate value theorem

Q.14 Consider the function f (x) =

0xfor0

0xforx

sinx

then the number of points in (0, 1) where the

derivative f (x) vanishes , is(A) 0 (B) 1 (C) 2 (D) infinite

Q.15 The sum of lengths of the hypotenuse and another side of a right angled triangle is given. The area of thetriangle will be maximum if the angle between them is :

(A)6

(B)4 (C)

3

(D)512

Q.16 In which of the following functions Rolle’s theorem is applicable?

(A) f(x) =

1x,0

1x0,x on [0, 1] (B) f(x) =

0x,0

0x,x

xsin

on [–, 0]

(C) f(x) =1x

6xx2

on [–2,3] (D) f(x) =

1xif6

]3,2[on,1xif1x

6x5x2x 23

Q.17 Suppose that f (0) = – 3 and f ' (x) 5 for all values of x. Then the largest value which f (2) can attain is(A) 7 (B) – 7 (C) 13 (D) 8

Q.18 The tangent to the graph of the function y = f(x) at the point with abscissa x = a forms with the x-axis anangle of /3 and at the point with abscissa x = b at an angle of /4, then the value of the integral,

a

b

f (x) . f (x) dx is equal to

(A) 1 (B) 0 (C) 3 (D) –1[ assume f (x) to be continuous ]

Q.19 Let C be the curve y = x3 (where x takes all real values). The tangent at A meets the curve again at B. If thegradient at B is K times the gradient at A then K is equal to

(A) 4 (B) 2 (C) – 2 (D) 1/4

Q.20 The vertices of a triangle are (0, 0), (x, cos x) and (sin3x, 0) where 0 < x <2

. The maximum area for sucha triangle in sq. units, is

(A)32

33(B)

323

(C) 324

(D)32

36

Q.21 The subnormal at any point on the curve xyn = an + 1 is constant for :(A) n = 0 (B) n = 1 (C) n = 2 (D) no value of n

Q.22 Equation of the line through the point (1/2, 2) and tangent to the parabola y = x2

2 + 2 and secant to the

curve y = 4 2 x is :(A) 2x + 2y 5 = 0 (B) 2x + 2y 3 = 0 (C) y 2 = 0 (D) none

Q.23 The lines y = 32

x and y = 25

x intersect the curve 3x2

+ 4xy + 5y2 4 = 0 at the points P and Q respectively.The tangents drawn to the curve at P and Q

(A) intersect each other at angle of 45º(B) are parallel to each other(C) are perpendicular to each other(D) none of these

Q.24 The least value of 'a' for which the equation,4 1

1sin sinx x

= a has atleast one solution on the interval (0, /2) is :

(A) 3 (B) 5 (C) 7 (D) 9

Q.25 If f(x) = 4x3 x2 2x + 1 and g(x) = [ Min f t t x xx x

( ) : ;;

0 0 13 1 2

then

g 14

+ g 34

+ g 54

has the value equal to :

(A)74 (B)

94 (C)

134 (D)

52

Q.26 Given : f (x) =3/2

x214

g (x) =

0x,1

0x,x

]x[nta

h (x) = {x} k (x) = )3x(log25

then in [0, 1] Lagranges Mean Value Theorem is NOT applicable to(A) f, g, h (B) h, k (C) f, g (D) g, h, k

Q.27 Two curves C1 : y = x2 – 3 and C2 : y = kx2 , Rk intersect each other at two different points. Thetangent drawn to C2 at one of the points of intersection A (a,y1) , (a > 0) meets C1 again at B(1,y2) 21 yy . The value of ‘a’ is

(A) 4 (B) 3 (C) 2 (D) 1

Q.28 f (x) = dxx1

1x1

1222

then f is

(A) increasing in (0,) and decreasing in (–, 0) (B) increasing in (–, 0) and decreasing in (0,(C) increasing in (– , (D) decreasing in (– ,

Q.29 The lower corner of a leaf in a book is folded over so as to just reach the inner edge of the page. Thefraction of width folded over if the area of the folded part is minimum is :

(A) 5/8 (B) 2/3 (C) 3/4 (D) 4/5Q.30 A rectangle with one side lying along the x-axis is to be inscribed in the closed region of the xy plane

bounded by the lines y = 0, y = 3x, and y = 30 – 2x. The largest area of such a rectangle is

(A) 8135

(B) 45 (C) 2135

(D) 90

Q.31 Which of the following statement is true for the function

0xx43x

1x0x

1xx

)x(f

3

3

(A) It is monotonic increasing Rx (B) f (x) fails to exist for 3 distinct real values of x(C) f (x) changes its sign twice as x varies from (– , )(D) function attains its extreme values at x1 & x2 , such that x1, x2 > 0

Q.32 A closed vessel tapers to a point both at its top E and its bottom F and is fixed with EF vertical when thedepth of the liquid in it is x cm, the volume of the liquid in it is, x2 (15 x) cu. cm. The length EF is:

(A) 7.5 cm (B) 8 cm (C) 10 cm (D) 12 cmQ.33 Coffee is draining from a conical filter, height and diameter both 15 cms into a cylinderical coffee pot diameter

15 cm. The rate at which coffee drains from the filter into the pot is 100 cu cm /min.The rate in cms/min at which the level in the pot is rising at the instant when the coffee in the pot is 10 cm, is

(A)16

9(B)

925

(C)35

(D)9

16

Q.34 Let f (x) and g (x) be two differentiable function in R and f (2) = 8, g (2) = 0, f (4) = 10 and g (4) = 8 then(A) g ' (x) > 4 f ' (x) x (2, 4) (B) 3g ' (x) = 4 f ' (x) for at least one x (2, 4)(C) g (x) > f (x) x (2, 4) (D) g ' (x) = 4 f ' (x) for at least one x (2, 4)

Q.35 Let m and n be odd integers such that o < m < n. If f(x) = xmn for x R, then

(A) f(x) is differentiable every where (B) f (0) exists(C) f increases on (0,) and decreases on (–, 0) (D) f increases on R

Q.36 A horse runs along a circle with a speed of 20 km/hr . A lantern is at the centre of the circle . A fence isalong the tangent to the circle at the point at which the horse starts . The speed with which the shadow ofthe horse move along the fence at the moment when it covers 1/8 of the circle in km/hr is

(A) 20 (B)40 (C) 30 (D) 60Q.37 Give the correct order of initials T or F for following statements. Use T if statement is true and F if it is

false.Statement-1: If f : R R and c R is such that f is increasing in (c – , c) and f is decreasing in(c, c +) then f has a local maximum at c. Where is a sufficiently small positive quantity.

Statement-2 : Let f : (a, b) R, c (a, b). Then f can not have both a local maximum and a point ofinflection at x = c.

Statement-3 : The function f (x) = x2 | x | is twice differentiable at x = 0.Statement-4 : Let f : [c – 1, c + 1] [a, b] be bijective map such that f is differentiable at c then f–1 is

also differentiable at f (c).(A) FFTF (B) TTFT (C) FTTF (D) TTTF

Q.38 Let f : [–1, 2] R be differentiable such that 0 f ' (t) 1 for t [–1, 0] and – 1 f ' (t) 0 fort [0, 2]. Then

(A) – 2 f (2) – f (–1) 1 (B) 1 f (2) – f (–1) 2(C) – 3 f (2) – f (–1) 0 (D) – 2 f (2) – f (–1) 0

Q.39 A curve is represented by the equations, x = sec2 t and y = cot t where t is a parameter. If the tangentat the point P on the curve where t = /4 meets the curve again at the point Q then PQ is equal to:

(A) 5 32

(B) 5 52

(C) 2 53

(D) 3 52

Q.40 For all a, b R the function f (x) = 3x4 4x3 + 6x2 + ax + b has :(A) no extremum (B) exactly one extremum(C) exactly two extremum (D) three extremum .

Q.41 The set of values of p for which the equation ln x px = 0 possess three distinct roots is

(A)

e1,0 (B) (0, 1) (C) (1,e) (D) (0,e)

Q.42 The sum of the terms of an infinitely decreasing geometric progression is equal to the greatest value of thefunction f (x) = x3 + 3x – 9 on the interval [– 2, 3]. If the difference between the first and the secondterm of the progression is equal to f ' (0) then the common ratio of the G.P. is

(A) 1/3 (B) 1/2 (C) 2/3 (D) 3/4Q.43 The lateral edge of a regular hexagonal pyramid is 1 cm. If the volume is maximum, then its height must be

equal to :

(A)13

(B)23

(C) 13

(D) 1

Q.44 The lateral edge of a regular rectangular pyramid is 'a' cm long . The lateral edge makes an angle withthe plane of the base. The value of for which the volume of the pyramid is greatest, is :

(A)4 (B) sin1 2

3(C) cot1 2 (D)

3

Q.45 In a regular triangular prism the distance from the centre of one base to one of the vertices of the otherbase is l. The altitude of the prism for which the volume is greatest :

(A) 2

(B)3

(C) 3

(D) 4

Q.46 Let f (x) =

1xif)2x(

1xifx

3

53

then the number of critical points on the graph of the function is(A) 1 (B) 2 (C) 3 (D) 4

Q.47 The curve y exy + x = 0 has a vertical tangent at :(A) (1, 1) (B) (0, 1) (C) (1, 0) (D) no point

Q.48 Number of roots of the equation x2 . e2 x = 1 is :(A) 2 (B) 4 (C) 6 (D) zero

Q.49 The point(s) at each of which the tangents to the curve y = x3 3x2 7x + 6 cut off on the positive semiaxis OX a line segment half that on the negative semi axis OY then the co-ordinates the point(s) is/aregiven by :

(A) ( 1, 9) (B) (3, 15) (C) (1, 3) (D) none

Q.50 A curve with equation of the form y = ax4 + bx3 + cx + d has zero gradient at the point (0, 1) and alsotouches the x-axis at the point (1, 0) then the values of x for which the curve has a negative gradient are

(A) x > 1 (B) x < 1 (C) x < 1 (D) 1 x 1Q.51 Number of solution(s) satisfying the equation, 3x2 2x3 = log2 (x2 + 1) log2 x is :

(A) 1 (B) 2 (C) 3 (D) noneQ.52 Consider the function

f (x) = x cos x – sin x, then identify the statement which is correct .(A) f is neither odd nor even (B) f is monotonic decreasing at x = 0(C) f has a maxima at x = (D) f has a minima at x = –

Q.53 Consider the two graphs y = 2x and x2 – xy + 2y2 = 28. The absolute value of the tangent of the anglebetween the two curves at the points where they meet, is

(A) 0 (B) 1/2 (C) 2 (D) 1

Q.54 The x-intercept of the tangent at any arbitrary point of the curve 22 yb

xa = 1 is proportional to:

(A) square of the abscissa of the point of tangency(B) square root of the abscissa of the point of tangency(C) cube of the abscissa of the point of tangency(D) cube root of the abscissa of the point of tangency.

Q.55 For the cubic, f (x) = 2x3 + 9x2 + 12x + 1 which one of the following statement, does not hold good?(A) f (x) is non monotonic(B) increasing in (–, – 2) (–1,) and decreasing is (–2, –1)(C) f : R R is bijective(D) Inflection point occurs at x = – 3/2

Q.56 The function 'f' is defined by f(x) = xp (1 x)q for all x R, where p,q are positive integers, has amaximum value, for x equal to :

(A) pqp q

(B) 1 (C) 0 (D)p

p q

Q.57 Let h be a twice continuously differentiable positive function on an open interval J. Letg(x) = ln )x(h for each x J

Suppose 2)x('h > h''(x) h(x) for each x J. Then(A) g is increasing on J (B) g is decreasing on J(C) g is concave up on J (D) g is concave down on J

Q.58 Let f (x) =

21xif0

21xif

1x2)1x6)(1x(

then at x = 21

(A) f has a local maxima (B) f has a local minima(C) f has an inflection point (D) f has a removable discontinuity

Q.59 Let f (x) and g (x) be two continuous functions defined from R R, such that f (x1) > f (x2) andg (x1) < g (x2), x1 > x2 , then solution set of )2(gf 2 > )43(gf is

(A) R (B) (C) (1, 4) (D) R – [1, 4]

Q.60 If f(x) =x

x2

(t 1) dt , 1 x 2, then global maximum value of f(x) is :

(A) 1 (B) 2 (C) 4 (D) 5

Q.61 A right triangle is drawn in a semicircle of radius 1/2 with one of its legs along the diameter. The maximumarea of the triangle is

(A) 41

(B)32

33(C)

1633

(D) 81

Q.62 At any two points of the curve represented parametrically by x = a (2 cos t cos 2t) ;y = a (2 sin t sin 2t) the tangents are parallel to the axis of x corresponding to the values of the parameter

t differing from each other by :(A) 2/3 (B) 3/4 (C) /2 (D) /3

Q.63 Let x be the length of one of the equal sides of an isosceles triangle, and let be the angle between them.If x is increasing at the rate (1/12) m/hr, and is increasing at the rate of /180 radians/hr then the ratein m2/hr at which the area of the triangle is increasing when x = 12 m and = /4

(A) 21/2

521 (B) 2

73 · 21/2 (C)

523 21

(D) 21/2

52

1

Q.64 If the function f (x) =4x

xx3t 2

, where 't' is a parameter has a minimum and a maximum then the range

of values of 't' is(A) (0, 4) (B) (0,) (C) (– , 4) (D) (4,)

Q.65 The least area of a circle circumscribing any right triangle of area S is :(A) S (B) 2 S (C) 2 S (D) 4 S

Q.66 A point is moving along the curve y3 = 27x. The interval in which the abscissa changes at slower rate than ordinate, is(A) (–3 , 3) (B) (– , ) (C) (–1, 1) (D) (– , –3) (3, )

Q.67 Let f (x) and g (x) are two function which are defined and differentiable for all x x0. If f (x0) = g (x0) andf ' (x) > g ' (x) for all x > x0 then

(A) f (x) < g (x) for some x > x0 (B) f (x) = g (x) for some x > x0(C) f (x) > g (x) only for some x > x0 (D) f (x) > g (x) for all x > x0

Q.68 P and Q are two points on a circle of centre C and radius , the angle PCQ being 2 then the radius of the circle inscribed in the triangle CPQis maximum when

(A)2213sin

(B)2

15sin (C)

215sin

(D)4

15sin

Q.69 The line which is parallel to x-axis and crosses the curve y = x at an angle of4

is

(A) y = 1/2 (B) x = 1/2 (C) y = 1/4 (D) y = 1/2

Q.70 The function S(x) =

x

0

2dt

2tsin has two critical points in the interval [1, 2.4]. One of the critical points

is a local minimum and the other is a local maximum. The local minimum occurs at x =

(A) 1 (B) 2 (C) 2 (D)2

Q.71 For a steamer the consumption of petrol (per hour) varies as the cube of its speed (in km). If the speed ofthe current is steady at C km/hr then the most economical speed of the steamer going against the currentwill be

(A) 1.25 C (B) 1.5 C (C) 1.75C (D) 2 CQ.72 Let f and g be increasing and decreasing functions, respectively from [0,) to [0,). Let h (x) = f [g

(x)] . If h (0) = 0, then h (x) h (1) is :(A) always zero (B) strictly increasing (C) always negative (D) always positive

Q.73 The set of value(s) of 'a' for which the function f (x) = a x3

3 + (a + 2) x2 + (a 1) x + 2 possess a negative

point of inflection .(A) (, 2) (0, ) (B) { 4/5 }(C) ( 2, 0) (D) empty set

Q.74 A function y = f (x) is given by x = 11 2 t

& y = 11 2t t( )

for all t > 0 then f is :

(A) increasing in (0, 3/2) & decreasing in (3/2,)(B) increasing in (0, 1)(C) increasing in (0,)(D) decreasing in (0, 1)

Q.75 The set of all values of 'a ' for which the function ,

f (x) = (a2 3 a + 2) cos sin2 2

4 4x x

+ (a 1) x + sin 1 does not possess critical points is:

(A) [1,) (B) (0, 1) (1, 4) (C) ( 2, 4) (D) (1, 3) (3, 5)Q.76 Read the following mathematical statements carefully:

I. Adifferentiable function ' f ' with maximum at x = c f ''(c) < 0.II. Antiderivative of a periodic function is also a periodic function.

III. If f has a period T then for any a R. T

0

dx)x(f = T

0

dx)ax(f

IV. If f (x) has a maxima at x = c , then 'f ' is increasing in (c – h, c) and decreasing in (c, c + h) ash 0 for h > 0.

Now indicate the correct alternative.(A) exactly one statement is correct. (B) exactly two statements are correct.(C) exactly three statements are correct. (D) All the four statements are correct.

Q.77 If the point of minima of the function, f(x) = 1 + a2x – x3 satisfy the inequality

x xx x

2

2

25 6

< 0, then 'a' must lie in the interval:

(A) 3 3 3 3, (B) 2 3 3 3,

(C) 2 3 3 3, (D) 3 3 2 3 2 3 3 3, ,

Q.78 The radius of a right circular cylinder increases at a constant rate. Its altitude is a linear function of theradius and increases three times as fast as radius. When the radius is 1cm the altitude is 6 cm. When theradius is 6cm, the volume is increasing at the rate of 1Cu cm/sec. When the radius is 36cm, the volumeis increasing at a rate of n cu. cm/sec. The value of 'n' is equal to:

(A) 12 (B) 22 (C) 30 (D) 33

Q.79 Two sides of a triangle are to have lengths 'a' cm & 'b' cm. If the triangle is to have the maximum area, thenthe length of the median from the vertex containing the sides 'a' and 'b' is

(A)12

2 2a b (B)23a b

(C)a b2 2

2

(D)a b 23

Q.80 Let a > 0 and f be continuous in [– a, a]. Suppose that f ' (x) exists and f ' (x) 1 for all x (– a, a). If f(a) = a and f (– a) = – a then f (0)

(A) equals 0 (B) equals 1/2 (C) equals 1 (D) is not possible to determineQ.81 The lines tangent to the curves y3 – x2y + 5y – 2x = 0 and x4 – x3y2 + 5x + 2y = 0 at the origin intersect

at an angle equal to(A) /6 (B) /4 (C) /3 (D) /2

Q.82 The cost function at American Gadget is C(x) = x3 – 6x2 + 15x (x in thousands of units and x > 0)The production level at which average cost is minimum is(A) 2 (B) 3 (C) 5 (D) none

Q.83 A rectangle has one side on the positive y-axis and one side on the positive x - axis. The upper right hand vertex on the curve y =nxx 2

. The

maximum area of the rectangle is(A) e–1 (B) e – ½ (C) 1 (D) e½

Q.84 A particle moves along the curve y = x3/2 in the first quadrant in such a way that its distance from the origin increases at the rate of 11 units persecond. The value of dx/dt when x = 3 is

(A) 4 (B)92

(C)3 32

(D) none

Q.85 Number of solution of the equation 3tanx + x3 = 2 in

4,0 is

(A) 0 (B) 1 (C) 2 (D) 3

Q.86 Let f (x) = ax2 – b | x |, where a and b are constants. Then at x = 0, f (x) has(A) a maxima whenever a > 0, b > 0 (B) a maxima whenever a > 0, b < 0(C) minima whenever a > 0, b > 0 (D) neither a maxima nor minima whenever a > 0, b < 0

Q.87 Let f (x) =

x

1

dtttn)t(nt ll (x > 1) then

(A) f (x) has one point of maxima and no point of minima.(B) f ' (x) has two distinct roots(C) f (x) has one point of minima and no point of maxima(D) f (x) is monotonic

Q.88 Consider f (x) = | 1 – x | 1 x 2 and

g (x) = f (x) + b sin2

x, 1 < x < 2

then which of the following is correct?(A) Rolles theorem is applicable to both f, g and b = 3/2(B) LMVT is not applicable to f and Rolles theorem if applicable to g with b = 1/2(C) LMVT is applicable to f and Rolles theorem is applicable to g with b = 1(D) Rolles theorem is not applicable to both f, g for any real b.

Q.89 Consider f (x) =

x

0

dtt1t and g (x) = f (x) for x

3,21

If P is a point on the curve y = g(x) such that the tangent to this curve at P is parallel to a chord joining the

points

21g,

21

and (3, g(3) ) of the curve, then the coordinates of the point P

(A) can't be found out (B)

2865,

47

(C) (1, 2) (D)

65,

23

Q.90 The co-ordinates of the point on the curve 9y2 = x3 where the normal to the curve makes equal interceptswith the axes is

(A)

31,1 (B) 3,3 (C)

38,4 (D)

56

52,

56

Q.91 The angle made by the tangent of the curve x = a (t + sint cost) ; y = a (1 + sint)2 with the x-axis at anypoint on it is

(A) t241

(B) tcostsin1

(C) t241

(D) t2costsin1

Q.92 If f (x) = 1 + x + x

1

2 dtnt2tn ll , then f (x) increases in

(A) (0,) (B) (0, e–2) (1, ) (C) no value (D) (1,)

Q.93 The function f (x) =)xe(n)x(n

ll

is :

(A) increasing on [0,) (B) decreasing on [0, )(C) increasing on [0, /e) & decreasing on [/e,) (D) decreasing on [0, /e) & increasing on [/e,)Directions for Q.94 to Q.96Suppose you do not know the function f (x), however some information about f (x) is listed below. Read

the following carefully before attempting the questions(i) f (x) is continuous and defined for all real numbers(ii) f '(–5) = 0 ; f '(2) is not defined and f '(4) = 0(iii) (–5, 12) is a point which lies on the graph of f (x)(iv) f ''(2) is undefined, but f ''(x) is negative everywhere else.(v) the signs of f '(x) is given below

Q.94 On the possible graph of y = f (x) we have(A) x = – 5 is a point of relative minima.(B) x = 2 is a point of relative maxima.(C) x = 4 is a point of relative minima.(D) graph of y = f (x) must have a geometrical sharp corner.

Q.95 From the possible graph of y = f (x), we can say that(A) There is exactly one point of inflection on the curve.(B) f (x) increases on – 5 < x < 2 and x > 4 and decreases on – < x < – 5 and 2 < x < 4.(C) The curve is always concave down.(D) Curve always concave up.

Q.96 Possible graph of y = f (x) is

(A) (B)

(C) (D)

Directions for Q.97 to Q.100

Consider the function f (x) =x

x11

then

Q.97 Domain of f (x) is(A) (–1, 0) (0, ) (B) R – { 0 } (C) (–, –1) (0, ) (D) (0,)

Q.98 Which one of the following limits tends to unity?

(A) )x(Limx

f

(B) )x(Lim0x

f

(C) )x(Lim1x

f

(D) )x(Limx

f

Q.99 The function f (x)(A) has a maxima but no minima (B) has a minima but no maxima(C) has exactly one maxima and one minima (D) has neither a maxima nor a minima

Q.100 Range of the function f (x) is(A) (0,) (B) (–, e) (C) (1, ) (D) (1, e) (e, )

Q.101 A cube of ice melts without changing shape at the uniform rate of 4 cm3/min. The rate of change ofthe surface area of the cube, in cm2/min, when the volume of the cube is 125 cm3, is

(A) – 4 (B) – 16/5 (C) – 16/6 (D) – 8/15Q.102 Let f (1) = – 2 and f ' (x) 4.2 for 1 x 6. The smallest possible value of f (6), is

(A) 9 (B) 12 (C) 15 (D) 19Q.103 Which of the following six statements are true about the cubic polynomial

P(x) = 2x3 + x2 + 3x – 2?(i) It has exactly one positive real root.(ii) It has either one or three negative roots.(iii) It has a root between 0 and 1.(iv) It must have exactly two real roots.(v) It has a negative root between – 2 and –1.(vi) It has no complex roots.(A) only (i), (iii) and (vi) (B) only (ii), (iii) and (iv)(C) only (i) and (iii) (D) only (iii), (iv) and (v)

Q.104 Given that f (x) is continuously differentiable on a x b where a < b, f (a) < 0 and f (b) > 0, whichof the following are always true?

(i) f (x) is bounded on a x b.(ii) The equation f (x) = 0 has at least one solution in a < x < b.(iii) The maximum and minimum values of f (x) on a x b occur at points where f ' (c) = 0.(iv) There is at least one point c with a < c < b where f ' (c) > 0.(v) There is at least one point d with a < d < b where f ' (c) < 0.(A) only (ii) and (iv) are true (B) all but (iii) are true(C) all but (v) are true (D) only (i), (ii) and (iv) are true

Q.105 Consider the function f (x) = 8x2 – 7x + 5 on the interval [–6, 6]. The value of c that satisfies theconclusion of the mean value theorem, is

(A) – 7/8 (B) – 4 (C) 7/8 (D) 0Q.106 Consider the curve represented parametrically by the equation

x = t3 – 4t2 – 3t andy = 2t2 + 3t – 5 where t R.

If H denotes the number of point on the curve where the tangent is horizontal and V the number of pointwhere the tangent is vertical then

(A) H = 2 and V = 1 (B) H = 1 and V = 2(C) H = 2 and V = 2 (D) H = 1 and V = 1

Q.107 At the point P(a, an) on the graph of y = xn (n N) in the first quadrant a normal is drawn. The

normal intersects the y-axis at the point (0, b). If21bLim

0a

, then n equals

(A) 1 (B) 3 (C) 2 (D) 4Q.108 Suppose that f is a polynomial of degree 3 and that f ''(x) 0 at any of the stationary point. Then

(A) f has exactly one stationary point. (B) f must have no stationary point.(C) f must have exactly 2 stationary points. (D) f has either 0 or 2 stationary points.

Q.109 Let f (x) =

0xfor8x

0xforx

2

2

. Then x intercept of the line that is tangent to the graph of f (x)

is(A) zero (B) – 1 (C) –2 (D) – 4

Q.110 Suppose that f is differentiable for all x and that f '(x) 2 for all x. If f (1) = 2 and f (4) = 8 thenf (2) has the value equal to

(A) 3 (B) 4 (C) 6 (D) 8Q.111 There are 50 apple trees in an orchard. Each tree produces 800 apples. For each additional tree

planted in the orchard, the output per additional tree drops by 10 apples. Number of trees that should beadded to the existing orchard for maximising the output of the trees, is

(A) 5 (B) 10 (C) 15 (D) 20

Q.112 The ordinate of all points on the curve y =xcos3xsin2

122

where the tangent is horizontal, is

(A) always equal to 1/2(B) always equal to 1/3(C) 1/2 or 1/3 according as n is an even or an odd integer.(D) 1/2 or 1/3 according as n is an odd or an even integer.

Select the correct alternatives : (More than one are correct)

Q.113 The equation of the tangent to the curvexa

yb

n n

= 2 (n N) at the point with abscissa equal

to 'a' can be :

(A) xa

yb

= 2 (B) xa

yb

= 2 (C) xa

yb

= 0 (D) xa

yb

= 0

Q.114 The function y = 2 12

xx

(x 2) :

(A) is its own inverse (B) decreases for all values of x(C) has a graph entirely above x-axis (D) is bound for all x.

Q.115 Ifxa

yb

= 1 is a tangent to the curve x = Kt, y =Kt

, K > 0 then :

(A) a > 0, b > 0 (B) a > 0, b < 0 (C) a < 0, b > 0 (D) a < 0, b < 0

Q.116 The extremum values of the function f(x) = 14

14sin cosx x

, where x R is :

(A)4

8 2(B)

2 28 2

(C)2 24 2 1

(D)4 28 2

Q.117 The function f(x) = 1 4

0

tx

dt is such that :

(A) it is defined on the interval [1, 1] (B) it is an increasing function(C) it is an odd function (D) the point (0, 0) is the point of inflection

Q.118 Let g(x) = 2 f (x/2) + f (1 x) and f (x) < 0 in 0 x 1 then g(x) :(A) decreases in [0, 2/3) (B) decreases in (2/3, 1](C) increases in [0, 2/3) (D) increases in (2/3, 1]

Q.119 The abscissa of the point on the curve xy = a + x, the tangent at which cuts off equal intersectsfrom the co-ordinate axes is : ( a > 0)

(A) a2

(B) a2

(C) a 2 (D) a 2

Q.120 The function sin ( )sin ( )

x ax b

has no maxima or minima if :

(A) b a = n , n I (B) b a = (2n + 1) , n I(C) b a = 2n , n I (D) none of these .

Q.121 The co-ordinates of the point P on the graph of the function y = e–|x| where the portion of thetangent intercepted between the co-ordinate axes has the greatest area, is

(A) 1 1,e

(B)

1 1,e (C) (e, e–e) (D) none

Q.122 Let f(x) = (x2 1)n (x2 + x + 1) then f(x) has local extremum at x = 1 when :(A) n = 2 (B) n = 3 (C) 4 (D) n = 6

Q.123 For the function f(x) = x4 (12 ln x 7)(A) the point (1,7) is the point of inflection (B) x = e1/3 is the point of minima(C) the graph is concave downwards in (0, 1) (D) the graph is concave upwards in (1,)

Q.124 The parabola y = x2 + px + q cuts the straight line y = 2x 3 at a point with abscissa 1. If thedistance between the vertex of the parabola and the xaxis is least then :

(A) p = 0 & q = 2(B) p = 2 & q = 0(C) least distance between the parabola and x axis is 2(D) least distance between the parabola and x axis is 1

Q.125 The co-ordinates of the point(s) on the graph of the function, f(x) = x x3 2

352

+ 7x - 4 where the

tangent drawn cut off intercepts from the co-ordinate axes which are equal in magnitude but opposite insign, is

(A) (2, 8/3) (B) (3, 7/2) (C) (1, 5/6) (D) none

Q.126 On which of the following intervals, the function x100 + sin x 1 is strictly increasing.(A) ( 1, 1) (B) (0, 1) (C) (/2, ) (D) (0, /2)

Q.127 Let f(x) = 8x3 – 6x2 – 2x + 1, then(A) f(x) = 0 has no root in (0,1) (B) f(x) = 0 has at least one root in (0,1)(C) f (c) vanishes for some )1,0(c (D) none

Q.128 Equation of a tangent to the curve y cot x = y3 tan x at the point where the abscissa is4

is :

(A) 4x + 2y = + 2 (B) 4x 2y = + 2 (C) x = 0 (D) y = 0

Q.129 Let h (x) = f (x) {f (x)}2 + {f (x)}3 for every real number ' x ' , then :(A) 'h ' is increasing whenever ' f ' is increasing(B) 'h ' is increasing whenever ' f ' is decreasing(C) ' h ' is decreasing whenever ' f ' is decreasing(D) nothing can be said in general.

Q.130 If the side of a triangle vary slightly in such a way that its circum radius remains constant, then,daA

dbB

d cCcos cos cos

is equal to:

(A) 6 R (B) 2R (C) 0 (D) 2R(dA + dB + dC)Q.131 In which of the following graphs x = c is the point of inflection .

(A) (B) (C) (D)

Q.132 An extremum value of y =0

x

(t 1) (t 2) dt is :

(A) 5/6 (B) 2/3 (C) 1 (D) 2

Q.1BQ.2DQ.3BQ.4BQ.5DQ.6CQ.7DQ.8AQ.9DQ.10CQ.11BQ.12AQ.13DQ.14DQ.15CQ.16DQ.17AQ.18DQ.19AQ.20AQ.21CQ.22AQ.23CQ.24DQ.25DQ.26AQ.27BQ.28CQ.29BQ.30CQ.31CQ.32CQ.33DQ.34DQ.35DQ.36BQ.37AQ.38AQ.39DQ.40BQ.41AQ.42CQ.43CQ.44CQ.45BQ.46CQ.47CQ.48BQ.49BQ.50CQ.51AQ.52BQ.53CQ.54CQ.55CQ.56DQ.57DQ.58CQ.59CQ.60CQ.61BQ.62AQ.63DQ.64CQ.65AQ.66CQ.67DQ.68BQ.69DQ.70CQ.71BQ.72AQ.73AQ.74BQ.75BQ.76AQ.77DQ.78DQ.79AQ.80AQ.81DQ.82BQ.83AQ.84AQ.85BQ.86AQ.87DQ.88CQ.89DQ.90CQ.91AQ.92AQ.93BQ.94DQ.95CQ.96CQ.97CQ.98BQ.99DQ.100DQ.101BQ.102DQ.103CQ.104DQ.105DQ.106BQ.107CQ.108DQ.109BQ.110BQ.111CQ.112DQ.113A,BQ.114A,BQ.115A,DQ.116A,CQ.117A,B,C,DQ.118B,C

Q.119A,BQ.120A,B,CQ.121A,BQ.122A,C,DQ.123A,B,C,DQ.124B,D

Q.125A,BQ.126B,C,DQ.127B,CQ.128A,B,DQ.129A,CQ.130C,D

Q.131A,B,DQ.132A,B

ANSWER KEY