Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 1 Section 5.3 Connecting f′ and f″ with the graph of f Applications of Derivatives Chapter 5

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 1

Section 5.3Connecting f′ and f″with the graph of f

Applications of DerivativesChapter 5


Quick Review

2

3

1/3

Factor the expression and use sign charts to solve the inequality.

1. 4 0

2. 9 0

Find the domain of and '.

3. ( ) 2

4. ( )

5. ( )1

x

x

x x

f f

f x xe

f x xxf xx


Quick Review

2

2

0.5

0.5

Find the horizontal assymptotes of the function’s graph.

6. ( ) 9

7. ( ) 9

3008. ( )1 10

3009. ( )2 10

x

x

x

x

f x x e

f x x e

f xe

f xe


Quick Review Solutions

2

3

1/3

2,2

3,0 (3

Factor the expression and use sign charts to solve the inequality.

1. 4 0

2. 9 0

Fin

, )

: all reals; ': all reals

: all r

d the domain of and '.

3. ( ) 2

4. ( ) al e

x f f

x

x x

f f

f

f

x xe

f x x

5. (

s;

)

': 0

: 1; 1

': 1xf x fx

f x

x f x


Quick Review Solutions

2

2

0.5

0.5

0

0

0 and 300

0 and

Find the horizontal assymptotes of the function’s graph.

6. ( ) 9

7. ( ) 9

3008. ( )

150

1 103009. ( )

2 10

x

x

x

x

f x x e

f x x e

f x

y

y

y y

y

e

fe

yx


What you’ll learn about Intervals of increase or decrease Both First and Second Derivative Tests for local extrema Intervals of upward or downward concavity Points of inflection Identification of key features of functions and their

derivatives …and whyDifferential calculus is a powerful problem-solving Tool precisely because of its usefulness for analyzing functions.


First Derivative Test for Local Extrema



The following test applies to a continuous function f(x).

1. If ' changes sign from positive to negative at , then has a local maximum value at .2. If ' changes sign from n

f c fc

f

At a critical point :c

egative to positive at , then has a local minimum value at .3. If ' does not change sign at , then has no local extreme value at .

If ' 0 ( ' 0) for , then has a loc

c fc

f c fc

f f x a f At a left endpoint :a

al maximum (minimum)value at .

If ' 0 ( ' 0) for , then has a local minimum (maximum)value at .

a

f f x b fb

At a right endpoint :b












Example Using the First Derivative Test

3

Use the First Derivative Test to find the local extreme values. Identify

any absolute extrema. ( ) 27 3f x x x

2

Since is differentiable for all real numbers, the only critical points are the zeros

of '. Solving '( ) 3 27 0, we find the zeros to be 3, and 3.

f

f f x x x x

The zeros partition the -axis into three intervals. Use a sign chart to find the sign on each interval. The First Derivative Test and the sign of ' tells us thatthere is a local maximum at 3 and

xf

x a local minimum at 3. The localmaximum value is ( 3) 57 and the local mimimum value is (3) 51.The range of ( ) is ( , ) so there is no absolute extrema.

xf f

f x


Concavity

The graph of a differentiable function ( ) is on an open interval if ' is increasing on .

on an open interval if ' is decreasing on .

y f xI y II y I

(a) concave up(b) concave down


Concavity


Concavity Test

The graph of a twice-differentiable function ( ) isconcave up on an open interval where " 0.concave down on an open interval where " 0.

y f xyy

(a) (b)


Example Determining Concavity

2Use the Concavity Test to determine the concavity of ( ) on the interval (2,8).

f x x

2Since " 2 is always positive, the graph of is concaveup on any interval. In particular, it is concave up on (2,8).

y y x


Point of Inflection

A point where the graph of a function has a tangent line and where the concavity changes is a point of inflection.


Example Finding Points of Inflection

2

Find all points of inflection of the graph of 2 .xy e

2

Find the second derivative of 2 .xy e

2 2

' 2 2 4x xy e x xe

2 2

" 4 4 2x xy e x e x

2 22 4 8x xe x e

2 2 4 1 2xe x

2 2The factor 4 is always positive. The factor 1 2 changes signxe x

1 1 2 1 2at . The points of inflection are , and , .2 2 2

xe e


Second Derivative Test for Local Extrema


Example Using the Second Derivative Test

3Find the local extreme values of ( ) 6 5.f x x x 2'( ) 3 6f x x

"( ) 6 .f x x

Test the critical points 2.x

" 2 6 2 0 has a local maximum at 2 andf f x

" 2 6 2 0 has a local minimum at 2.f f x


Learning about Functions from Derivatives


Quick Quiz for Sections 5.1 – 5.3

2 4

You should solve these problems without using a graphing calculator.

1. How many critical points does the function ( ) – 2 3 have?(A) One(B) Two(C) Three(D) Five(E) Nine

f x x x



2 4

You should solve these problems without using a graphing calculator.

1. How many critical points does the function ( ) 2 3 hav

(C)

e?(A) One(B) Two

(D) Five(E) Nine

Three

f x x x



22. For what value of does the function ( ) 2 3 have a relative maximum?(A) 3(B) 7 / 3(C) 5 / 2(D) 7/3(E) 5/2

x f x x x



22. For what value of does the function ( ) 2 3 have a relative maximum?(A) 3(B) 7 / 3(C) (D)

5 / 2

(E) 5 7/3

/2

x f x x x



2

3. If is a differentiable function such that ( ) 0 for all real

numbers , and if '( ) 9 ( ), which of the following

is true?(A) has a relative maximum at 3 and a relative minimumat 3.(B

g g x

x f x x g x

f xx

) has a relative minimum at 3 and a relative maximumat 3.(C) has a relative minima at 3 and at 3.(D) has a relative maxima at 3 and at 3.(E) It cannot be determined if has an

f xxf x xf x x

f

y relative extrema.



2

3. If is a differentiable function such that ( ) 0 for all real

numbers , and if '( ) 9 ( ), which of the following

is true?(A) has a relative maximum at 3 and a relativ

(

e minimum

Bat 3.

g g x

x f x x g x

f xx

(C) has a relative minima at 3 and at 3.(D) has a rela

) has a relative minimum at 3 and a relative maximu

tive maxima at 3 and at 3.(E) It cannot be determined if has

mat 3

an

.f x xf x

f

f xx

x

y relative extrema.

Documents

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 1 Section 5.3 Connecting f′ and f″ with the graph of f Applications of Derivatives Chapter 5