e2y d 3 d 3

If - - =0 at x = a, then find ---:!::: . If ---:!::: ~ 0 at x = a, dx2 dx3 dx3

, 3

then neither maximum nor minimum at x = a. If d Y = 0 dx3

h ' f' d d 4 Y If d 4 Y . . ' th atx = a,t en In --. - -> OI.e.+lveat x=a, en y dx4 dx 4

. .. d 'f d4y . . h IS ml111mum at x = a an 1 - - <0 I.e. -lve at x = a, t en

dx4

y is max. at x = a and so on. Parametric Form of a Function. Let a function

y = f (x) be represented in parametric form by the equations

x=~(t), y='I'(t). where ~(t) and 'I' (t) have derivatives both of first and second orders within a certain interval of t, and ~' (t) = O. Let att =t o''I'' (t) =0. Then

(a) if'l''' (to) <O,f (x) has a maximum at x = Xo = q> (t 0)

(b) if'l''' (t 0) >0 the functionf (x) has a mini- mum at x = x o = q>(to)

(c) if '1'" (t 0) =0, the question of the existence of an extreme value remains open.

Note: Sometimes we also use the term extreme values for mamimum or minimum values . The points at which f ' (x) does not exist are called critIcal points. If f' (x) =0, we say that y = f (x) is stationary at x = a.

Important Instructions 1. According to the given conditions of the problem

determine the function whose max. and min. v alues are to 'be found.

2. It may happen that the above function is not of single variable but contains more than one variable. In such cases there will be given certain other relation between these variables with the help of which by elimination we shall be able to reduce the above function to be of single variable. A function of the form

k f (x), k + f tx), [f (x)]k or [f (X)]l /k ,

where k is a + ive constant will be max. or min. according as f (x) is maximum or minimum provided f (x) >0.

Also if y is max. and min. then log y = z is also max. and min. provided y >0. (Note)

3.

Also Y = f (x ) is max. or min . according as z = 1 / f (x ) is min. or max. Max. and Min. occur alternately. Commit to Memory : In usual notations

(i) Area of a square= x 2 ,its perimeter = 4x. . I

(ii) Area of rectangle = xy, perimeter = 2 (x + y ). I

, ;

I

(iii) Area of a trapezium =! (sum of parallel sid es) 2

x distance between tl 1em. (iv) Area of a circle = nr2 , Perimeter = 2nr

(v) Volume of Sphere =.i nr3 its Surface = 47 cr2 3

(vi) Volume of a right cone =! nr2 h. 3

(vii) Toltal Surface of cone = nr (1' + 1) (viii) Its curved surface is nr10n ly.

(ix) Volume of a cylinder =n/;h.

(x) Total Surface =2nr (r + h), . (xi) Its curved surface is 2nrh.

(xii) Volume of a cuboid = xy z (xiii) Surface of a cuboid =2; (xy + yz ·t- zxJ . Greatest and lea!~t values IJf a fund ion. The reader

will do well to bear ):n m ind that a Ill8tXimum value of • f (x) at x=xo in an interval [a ,b] does not m:ean that it is greatest value of f (x ) in tha't interval. Thf6te may be a value of the function greater I,han a max i mum value. As a matter of fact, there may ex ist a miniI llum value of the funcion which is greater tha n or equal i to some maximum value of the function in [a, b] .

However, if a function f (x) is con'tinuous in a closed interval [a,b], then great.est (least) value of f (x) is attained at critical pointE;, or at the end points of the interval. Thus to find thee greatest (least) value of the function, we have to compute its valu e at all the critical points on the interval [a, b] and also the values f (a) , f (b) and choose the greatest (least) one out of the numbers thus obtained. If the interval is no t closed it may have neither the greatest nor the least vaJue.

Sign of f (x) for small values of x. The sign of f (x) is ' governed by the lowest degree terms in f (x). For example, for small x the sign of _.3.+ 4x + 7x2 will be governed by the constant term and hence - ive. Similarly the sign of x 3 -7 x11 + 20 x 15 will be governed by x 3 and so its sigr will be + ive or - ive according as x is + ive or - ive.

Maxima, minima of algebraic functions.

1. (a) If f (x) = x 5 - 5x4 + 5x 3 - 10 has local max. and

min. at x = p and x =q resp., then (p, q) =

(a) (0, 1)

(c) (1 , 0) (b) (l. (d) •

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2.

3.

4.

5.

6.

7.

8.

9.

10.

(b) The function f (x) = ~ + lS. has a local minimum x 2

at X =

(a) -2 (c) 1

(b) 0 (d) 2 (AIEEE 2006)

(e) Suppose the cubic x3 - px + q has tllree distinct

real roots where p >0 and q >0. Prove that the

cubic has minima at ~ and maxima at - H. (AIEEE 2008)

n

If a; E R then the expression L ) 2 • ai assumes Its i =

least value at x =

(a) '[.aj (b) 2La j (c) ( 1 (d) - La· n 1

The real number x when added 't it~ inverse gives the minimum value of the sum at x ~~l to (a) -2 (b) - 1 (c) 1 ( (d) 2

(A.I.E.E.E. 2003)

. x2 +x+1 The max. and mm. value of are

x2 -x+ 1

(a) (2, 1) (c) (1, 0) (d) (3 , 1)

2

For all real x, the minimum value of 1- x + x is ' . 1 + x+ x 2

(a) 0 (b) .1 / 3 (c) 1 (d) 3 (M.N.R. 1992)

• r x2 -3x+ 2. The functlOn J (x) = 2 IS equal to

Y + 2x-3

(a) min. at x =- 3, max. at x =1

(b) max. at x =-3

(c) Increasing in its domain (d) Decreasing in its domain

The maximum value of (log x) / x is (a) 1 (b) 2 / e (c) e

XX has a stationary point at .

(a) x =e (b) x =1 / e (c) x=1

(d) 1/ e

(d) x =..Je (Karnataka 1993)

or When x is positive, the minimum value of XX is

(a) e - 1 (b) e - l /e (c) e l le (d) ee

Greatest value of (1 / x) X is

(a) e (b) (e) lie

(c) (1/e)e (d) None of these (P.E.T. Raj. 1991)

The ma~imum value Of(~yx2 is

(a) 1 (b) e (c) el/e (d) None

Objective

11. The minimum value of ax + by when xy = r2 is

(a) 2r.JQb (b) 2ab.Jr (c) -2r.JQb (d) none

12. Given that f (x) = xl i x, x >0 has the maximum value

at x =e, then (a) ell >rce (b) e ll <rce (c) ell = rce (d) ell ~ rce

13. On the interval [0, 1] the function X25

(1 _ X)75 takes

the maximum value at the point

14.

(a) ~ (b) ~ (c) .! (d) 0 234

[I.LT. 1995]

In a sub-marine telegraph-cable the speed of signalling varies as x2 log (1 / x) where x is the ratio of the radius

of the case to that of covering. The greatest speed is attained when this ratio is (a) 1:e (b) 1:Je (c) 1:eJe (d) none

15. If y = a log I x I + bx2 + x has its extremum

values at x = -1 and x = 2, then (a) a=2,b=-1 (b) a =2,b= - 1/2

(c) 0 = -2,b = 1 / 2 (d) None uf these

16. (a) The function f (x) = X 4 - 62x2 + ax + 9 attains its

maximum value on the interval [0, 2] at x = 1. Then the value of a is (a) 120 (c) 52

(b) The function

(b) - 120 (d) none of these

f(x)=2x 3 - 9ax2 +12a2 x+l=0 has a local

maximum at x = (1 and a local minimum at x = ~ such that ~ = (1 2 then a is equal to :

(a) 0 (b) 1/4 (c) 2 (d) either 0 or 2

17. (a) Letj(x) =(1+b 2 )x2 +2bx+ 1 and letm(b) be the

minimum value of f (x). As b varies, the range of m (b) is

(a) [0,1] (b) Jo , ~J (c) 1~,lJ (d) ]0,1]

L2 (LLT. Sc. 2001) . 3 2 '

(b) Iff (x) =2x - 3x -12x+50n[-2,4]thenrelative

maximum occurs at x = (a) - 2 (c) 2

(b) - 1

(d) 4 (AIEEE 2002)

18. If minimum value of f (x) = ... i + 2bx + 2[;2 is greater

than maximum value of g (x) = - x2 -7fx+b 2 then for

x is real

(a) Icl>lblvlz (c) 0 <c <.Jib

(b) Iclvlz >b

(d) no real value of a. (Screening 2003)

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19. If the function f (x) = 2x3 - 9ax2 + 12a2 x + 1, where

a> 0, attains its max. and min. at p and q respectively such that p2 =q then a equals

(a) 1 (b) 1 .(e) 2 (d) 3 2

(A.I.E.E.E.2003)

20. For the curve y = xe x, the point

(a) x = -1 is a point of minima (b) x =0 is a point of minima (c) x = -1 is a point of maxima (d) x =0 is a point of maxima (M.N.R. 1990)

21. Divide 20 into two parts such that the product of one part and the cube of the other is maximum. The two parts are (a) (10,10) (b) (12,8) (c) (15, 5) (d) None

22. Divide 64 into two parts such that the sum of the cubes of two parts is minimum. The two parts are (a) 44,20 (b) 16,48 (c) 32,32 (d) 50,14

23. A particle is moving in a straight line such that its distance at any timj3 t is given by

4

X =~ _2t 3 + 4t 2 + 7. Then 4

(a) velocity is max. att =(6 -2"3) / 3 (b) acceleration is min. at t = 2 (c) min. distance is att =0,4 (d) None of these

24. The points on the curve 5x2 -8xy + 5y2 =4 whose

distance from the origin is maximum or minimum are (a) (../2,../2) (b) (-../2,-../2)

(c) (~ ,-~) (d) (- ~, ~) 25. The longest distance of the point (a, 0) from the curve

2x2 + y2 -2x=0 is given by

(a) ~1 + 2a + 2a2 (b) ~1 +2a-a2

(c) ~1-2a + 2a2 (d) ~1-2a+ a2

26. The point on the curve y = x / (1 + x 2) where the

tangent to the curve has greatest slope is

(a) (0 , 0) (b) (.13, ~)

(c) [-.13, -:) (d) none

27. The co-ordinates of a point of the parabola y = x2 + 7x + 2 which is closest to the straight line

y =3x-3is (a) (-2,8)" (b) (-2,-8) (c) {2,-8) (d) none

28. The point (0, 5) is closest to the curve x2 = 2y at

(a) (2"2,0) (b) (0,0)

(c) (2 , 2) (d) None of these 2 2

29. Find the point of the hyperbola £ - L = 1 24 18

which is nearest to the line 3x + 2y + 1 =0. Compute the distance between the point and the line.

(J.E.E. W.B. 1992)

30. The maximum distance of the point (k,O) from the curve 2X2 + y2 - 2x = ° is equal to

(a) ~1 + 2k -e (b) ~r-1-_2-k-+-2-k-2

(c) ~1 + 2k + 2k2 (d) ~1-2k + k 2

31. The co-ordinates of a point on the parabola y2 = 8x

whose distance from the circle x2 + (y + 6) 2 = 1 is minimum is (a) (2,4) (b) (2,-4) (c) (18,~12) (d) (8,8)

32. Two towns A and B are 60 km. apart. A school is to be built to serve 150 students in town A and 50 students in town B. If the total distance to be travelled by 200 students is to be as small as possible, then the school should be built at . (a) town B (b) 45 km. from town A

(c) town A (d) 45 km. from town B 33. The fuel charges for running a train are proportional to •

the square of the speed generated in miles per hour and costs Rs. 48 per hour at 16 miles per hour. The most economical speed if the fixed charges i.e. salaries etc. amount to Rs. 300 per hour. (a) 10 (b) 20 (c) 30 (d) 40

34. Petrol burnt in driving a motor boat varies as the cube of its velocity. The most economical speed when going against a current of e miles per hour in m.p.h. is (a)e (b)e/2 (e)3c / 2 (d) none

35. (a) Ifafunetionf (x)hasf ' (a) =Oandf" (a) =O,then (a) x=ais a maximum for f (x)

(b) x = a is a minimum for f (x)

(c) It is difficult to say (a) and (b) (d) f (x) is necessarily a constant function

(b) Iff" (x) <o,x E (a,b),f (e) attains maximum value at (e,f (e)), where a <c <b, then f ' (c) is equal to : (a) f(b)-f(a)

2

(b) f' (c) =0

(c) !'(c) f(a)-f (b) 2

(d) f' (c) =2(/ (b) - f (a)) 2 4 2n b 36. Let P(x)=aO +a1 x +a2x + ... ... +anx " e a

polynomial in a real variable x with 0< ao < a1 < az < ..... < an · The function P (x) has (a) Neither a max. nor a min. (b) Only one ~aximum

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(c) Only one minimum (d) None of these

37. The largest value of 2x3 - 3x2 -12x + 5 for

- 2 ::; x ::; 4 occurs at x = (a) -2 (b) -1 (c) 2 (d) 4

(M.N.R. 1993)

38. (a) The difference between the greatest and least values of the function f (x) = sin 2x - x on [- 1t / 2, 1t / 2] is

(a) 1t (b) J3 + 2! 2 3

() J3+..f2 1t (d) ~(J3+..f2) c +-2 6 2

(b) The difference between the greatest and the least values of the function f(x)=.b (at 2 +1+cost)dt,a>0,xE[2,3]is:

(a) 19a+l+sin3-sin2 3

(b) 18 a + 1 + 2 sin 3 3

(c) 18 a-l+2sin3 3

(d) none

(c) If p and q are positive real numbers such that p2 + q2 = 1, then the maximum value of(p + q) is

(a) 2 (b) ~ 2

1 (d) ..f2 (c) -..f2 (AIEEE 2007)

39. The function f(x)=r {2(t-l)(t-2)3 +3(t-l)2 (t -2)2}dt

attains its maximum at x = (a) 1 (b) 2 (c) 3 . (d) 4

40. The function f (x) = L~ t (e t -1) (t -1) (t _2)3 (t _3)5 dt

has a local minimum at x = (a) 0 (b) 1 tc) 2 (d) 3

41. The points of extremum of the function .

~(x)=ix e- t/2 (1-t 2 )dtare

(I . LT. 1999)

(a) x=o (b) x=l (c) x=1/2 (d) x=-l

42. (a) f(x)=1x t(t-l)(t-2)dttakesonits

minimum value at x = (a) 0 (b) 1 (c) 2 (d) none

{

X o <x:S;l (b) Let f (x) = 2 - e x - 1 : 1 < x ::; 2

x-e ,2<x::;3

and g'(x) =.b f (t) dt , then:

43.

44.

45.

46.

(a) g (x) has local minima at x = e and local maxima at x =l + In 2

(b) g (x) has local maxima, at x = 1 and local minima at x=2

(c) g (x) does not have local maxima (d) g (x) does not have local minima (I. LT. 2006)

(a) A cubic f (x) vanishes at x=-2 and has relative minimum/maximum at x = 1/ 3 such that

L\ f (x) dx = 14 . Then f (x) is equal to 3 .

(a) x3 + x2 - x (b) x3 + x2

- X + 1

(c) x 3 +x2 -x+2 (d) x 3 +X2 - x -2

(b) Let three degree polynomial function f (x) has local maximum at x = -1 and f (-1) = 2, f (3) = 18, f' (x) has a minima at x= O, then: (a) the distance between (-1,2) and (aJ (a))

where a denotes point where function has local max/min is 2.J5

(b) the function decreases from 1 to 2.J5

(c) the function increases from 1 to 2.J5

( d) the function decreases from -1 to 1 • (I.LT.2006)

The minimum value of ax + by where xy = c2 is

(a) 2c JOb (b) 2ab JC (c) -2cJOb (d) none of these

If a2 x4 + b 2 y4 =c 6 the maximum value of xy is

3 3 3 3 (a) _c_ (b) _c_ (c) ~ (d) ~

JOb .J2ab ab 2ab

The minimum value of f (x) = 13 - x 1 + 12 + xl + 15 - x 1 is (a) 7 (b) 10 (c) 8 (d) 0

47. (a) The interval into which the function

48.

y . 2 x -1 transforms the entire real line is x -3x+-3 _

(b) [- 1 / 3, 1] (a) [1/3, 2]

(c) [-1 / 3,2] ( d) none of these

(b) The set of values of A. for which the function

f (x) = (4A. - 3) (x + log 5) + 2 (A. -7). cot l! sin 2 l! 2 2

does not possess critical point is : (a) [1,00) (b) (2,00)

(c) (- 00, - 4 / 3) (d) (- 00, -1)

Letf(X)= {lx l for 0 <lxl ::; 2 . 1 for x =O.

Then at x = OJ has (a) a local maximum

(c) a local minimum

(b) no local maximum

(d) no extremum (!.I.T. Sc. 20(0)

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and Normals

(a) !!. 7

(b) ~ 7

(c) ~ 7

(d) ~ 7

49. Let(h,k) be a fixed point, where h >O,k >O.A straight line passing through this point cuts the positive direction of the comdinate axes at the points P and Q. Which of the following is the minimu:r:n area of the triangle OPQ, 0 being the origin

52. Iff (x) = (xz _l)n+l (xz +x +1),nEN and f (x) has a

local extreamm at x = 1, then n =

(a) hk (b) 2hk (a) 2 (b) 3 (c) 4 (d) 5

(~) . 1.hk (d) none of these TRU€ ANO FAlSE 2

50. The minimum value of 2(X2 -3'f +Z7 is

(!.I .T.1995) 1. A curve passes through the point (2, 0) and its gradient at the point( x, y) is xZ

- 2x for all values of x, then the

(a) 2Z7 (b) 2 point of maximum ordinate on the curve is ( 0, ~) . (c) 1 (d) none of these

51. If f(x)=x3 +3(a-7)xz +3(az - 9)x-2 where a>O,

has + ive poirit of maximum then a varies over an interval of length

MULTIPLE'GH01CE QUESTIONS' ..; :~ .

1 . . (ar(b) (b) (d)! ,2: .. (d) :4" (b}l;

9. (B) 16. (c) , U. (a) 13. (c)' ( (l)')" 18. !(a)

" ,~ .... 17. (a) (d) «l' (b) 20. (a)

26. (ic!) 2?- (ct 28. (d) ;. 30. (b)

35. (a) (c) (b) () ";' 36. (c) 37. (d) 38. (a) (a) ~ ... t': .

42. (a) (a, c) . (a, b) 43: (a) ' (c) 41. (b, dJ ,:~~ (b)

47. (a) (b) :;': ,

£l!) (b;S) " 48.1' (d) 49. ,(b) 50; (cy

TRUE AND FALSE

2. Iff' (x) =(x_a)zn (x_b)zm+l then

(i) x = a is a point of neither max. nor min. (ii) x =b is a point of maxima.

5. (b) 6. (c) 7. (d) 8. (b)

14. (b) 15. (b) 16. (a) (a) (b) (c) .

21. (c) 22. (c) 23. (a, b, c) , 24. (a,b,c,d) 25.

31. (b) 32. (c) 33. (d) 34. (c)

(b) (a) (e) (d) 39. (a) (a) 40. (b, d)

(b) (a) 44. (a) 45. (b) 46. (a)

51. (a) 52. (b, d)

(c)

~~~~·~.m~~i~ __ ~~~'-~~ __ ~'~~~~~~-'W_~~~~~~

ints/Sollitr ns~·to P~oblem Set (A] ~~~.a~::.":W:I:~:~:W::'!MI7"I"'W'=,,:§:w;:'"4:-WJ.-i:"";:£J~=':::"""""""""'''''''-''''''''''---~~!!:.oow--'''''''''o::m.""", ............... ~-~;

~i . .;.,:;~~€HQrcE . QUE$TIOI"IS . :.it>"ff!!tj:"" (b) Ans. (d). dy = _ ~ + 1. dx x Z 2 1. (a) Ans. (b).

dy 2 ( ) ( . ) 0 3 dy = 0 ~ xZ - 4 = 0 :. x = 2, - 2 - =5x x - l x-3 =0 : . X= ,1, dx

dx . d Z yz d

Z

y =~ = - ive at x = 2 and hence minimum. - = 10x(2x -6'X+3) =+ive,90 dx2 x 3

dxz (c) y = x 3 - px+q

at x = 3 :. Min. fp fp =-ive,-10 atx=l .. Max : =3xz - p~ x=V~,- V~ =0 at x =0 : . Neither

2ndMethod. Changeofsign:5xz isalways+ive dZy =6x=+iveatx = IE :. min. dx2 V:3

dy. dy . H x<l,~ =+ lve,x>l,~ = -lve dx dx = - iveat X=- ~ .. max.

Max. at x=l d ' dy 2. Ans. (d).

x <3,2 =- lve, x > 3, - = + lve dx dx

n

Min. at x=3 i =l

dy. · dy. x<O, - =+lve,x>O,- = +lve

dx dx dy La · - = 2nx-2La , =0 : . X= _l ~ (d).

.dx 1 n

No change and hence neither max. nor min.

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