3.3 Increasing and Decreasing Functions and the First Derivative Test.

3.3 Increasing and Decreasing Functions and the First Derivative Test

Test for Increasing and Decreasing FunctionsLet be a continuous function on [ , ] and differentiable on (a,b).f a b

1. '( ) 0 in ( , ) is increasing on [ , ].f x x a b f a b

2. '( ) 0 in ( , ) is decreasing on [ , ].f x x a b f a b

3. '( ) 0 in ( , ) is constant on [ , ].f x x a b f a b


x a x b

constant

decreasing increasing

' 0f x ' 0f x ' 0f x x

y

f

Guidelines to Finding Intervals on Which

a Function Is Increasing or Decreasing

Let be continuous on ( , ).f a b

1. Locate the critical numbers in ( , ) and determine test intervals.a b

2. Find the sign of '( ) at a test value in each interval.f x

3. Use the Test for Increasing and Decreasing Functions

to decide whether is increasing or decreasing on each interval.f


Without a calculator, find the open intervals on which3 23

( )2

f x x x

is increasing or decreasing.

Critical Numbers will give us

the location of possible extrema.

2'( ) 3 3f x x x 3 ( 1)x x

0,1 are the critical numbers.c

0


0 and 1 arBecause ' e the onl is defined 's, y critical numb ,ersf R

We simply need to create a table to test the three intervals

determined by the two critical numbers.

Interval 0x 0 1x 1 x Test Value 1x 1/ 2x 2x Sign of '( )f x positive negative positive

Conclusion Increasing Decreasing Increasing

2'( ) 3 3f x x x ( ) f x is


Graph

2'( ) 3 3f x x x


decreasing

increasing

increasing

1

1

11,

2

0,0

3 23( )

2f x x x

Test Points

is strictly monotonic on [a,b] if it is increasing or decreasing on the entire interval.f


3( )f x x( )f x { 0,0 1x

2 , 0x x

21 , 1x x

1

The First Derivative TestLet be a critical number of , which is continuous on

containing .

c f I

c

is differentiable on (except possibly at )f I c

( ) can be classified as follows.f c


'( ) changes from negative to positive at

( ) is a relative minim

1

um of

.

.

f x c

f c f

Relative Minimum

'( ) changes from positive to negative at

( ) is a relative maxim

2

um of

.

.

f x c

f c f

Relative Maximum

Neither relative minimum nor relative maximum

Applying the First Derivative Test

1Find the relative extrema of ( ) sin on (0,2π).

2f x x x

Critical Numbers?

1'( ) cos 0

2f x x

1 5cos ,

2 3 3x x

(since ' is defined 's, these are the only C 's) Vf R


Interval

Test Value

Sign of '( )f x

Conclusion IncreasingDecreasing

0,3

5,

3 3

5,2

3

/ 4x x 7 / 4x

positivenegative negative

Decreasing

By applying the First Derivative Test,

you can conclude that has a relative

minimum at

f

3x

and a relative maximum at3

.5

x

1'( ) cos

2f x x

( ) f x is


3x

5

3x

0,2

3

1

1sin

2y x x

Relative Min

Relative Max

5

3

5,?

3

,?3

3 3

3 6

3

6 2f

5 35 5 3

6

3

3 6 2f

3 3

6

5 3 3

6

22 3

Find the relative extrema of

( ) ( 4) . f x x

2 1/3

1/32

2'( ) ( 4) 2

34

3 4

Critical #'s @ 0, 2

f x x x

x

x


IncreasingDecreasing

0 0

, 2 2,0 0,2.INT

TV'( )f x

Decreasing Increasing

3x 1x 1x 3x 2,

'( 3)f '( 1)f '(1)f 0 '(3)f 0

( 2,0)

Relative Min at (2, (2))f (2,0)

Relative Min at ( 2, ( 2))f

Relative Max at (0, (0))f (0, 2.520)

2/32( ) 4f x x


( ) f x is

1/32

4'( )

3 4

xf x

x

0, 2x ,


2/32 4y x

4

2

Find the relative extrema of

1( ) .

xf x

x

2 2( )f x x x

3

22'( )f x x

x

3

1'( ) 2f x x

x


3

4

21

x

x

2 2

3

2( 1)( 1)x x

x

'( ) 0 at 1f x x '( ) at 0f x DNE x

3

1'( ) 2

xf x x


IncreasingDecreasing

0 0

, 1 1,0 0,1.INT

TV'( )f x

Decreasing Increasing

2x 1/ 2x 1/ 2x 2x 1,

'( 2)f '( 1/ 2)f '(1/ 2)f 0 '(2)f 0

( 1,2)

(0) DNEf

Relative Min at (1, (1))f (1,2)

Relative Min at ( 1, ( 1))f

2 2

3

2( 1)( 1)'

x xf x

x

4

2

1( )

xf x

x

0, 1x ,

The path of a projectile that is propelled at an angle is

y heighthorizontal distancex

g acceleration due to gravity

0 v initial velocity

22

20

sec(tan ) , 0

22

gy x x h

v


32 feet per second per secondg

0 24 feet per secondv

9 feeth What value of will produce

a maximum horizontal distance?

Write in terms of and find the max.x


22

20

sec(tan )

2

gy x x h

v

32

24

9 0

We may need Bell’s THM!!

22

20

sec(tan )

2

gy x x h

v

Write in terms of and find the max.x 3.3 Increasing and Decreasing Functions and the First Derivative Test

9 0

2

2secta 9

360 nx x

32

24

22

2

sectan tan 4 (9)

36

sec2

36

x

2

2

2tan tan sec

sec /18x


22

2

4sec

tan 936

236

tan ( )

secx

Tangent and secant!!! You must be kidding. Let’s write everything in terms of sine and cosine.

2 2

2 2

18 tan 18 tan sec

sec secx

218cos (sin sin 1)x

2 2

2 2

18 tan 18 tan sec

sec secx

22 2

2

sin sin 118cos 18cos

cos cosx

218sin cos 18cos 1 sinx

0x

Now find the which produces a maximum value of .x


Using the first derivative test would be very tedious.

Let's use technology to find / 0.dx d

35.264

Graph

and find where it equals 0.

dx

d

218cos sin sin 1x

218cos sin sin 1x 0 at 35.3 dx

d

HW 3.3 pp. 176-179/ 10,11,14,15,17,19,23-37odd,41,50-52,55


3.3 Increasing and Decreasing Functions and the First Derivative Test.

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