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OBJECTIVES:

1. Determine intervals on which a function is concave upward or concave downward.

2. Find any points of inflection of the graph of a function.

3. Apply the Second Derivative Test to find the relative extrema of a function.

1. Determine intervals on which a function is concave upward or concave downward.

2. Find any points of inflection of the graph of a function.

3. Apply the Second Derivative Test to find the relative extrema of a function.

1.
Applications of Differentiation

Copyright © Cengage Learning. All rights reserved.

Copyright © Cengage Learning. All rights reserved.

2.
Concavity and the Second

Derivative Test

Copyright © Cengage Learning. All rights reserved.

Derivative Test

Copyright © Cengage Learning. All rights reserved.

3.
Determine intervals on which a function is concave

upward or concave downward.

Find any points of inflection of the graph of a function.

Apply the Second Derivative Test to find relative

extrema of a function.

3

upward or concave downward.

Find any points of inflection of the graph of a function.

Apply the Second Derivative Test to find relative

extrema of a function.

3

4.
4

5.
You have seen that locating the intervals in which a function

f increases or decreases helps to describe its graph. In this

section, you will see how locating the intervals f' increases

or decreases can be used to determine where the graph of f

is curving upward or curving downward.

5

f increases or decreases helps to describe its graph. In this

section, you will see how locating the intervals f' increases

or decreases can be used to determine where the graph of f

is curving upward or curving downward.

5

6.
The following graphical interpretation of concavity is useful.

1. Let f be differentiable on an open interval I. If the graph

of f is concave upward on I, then the graph of f lies

above all of its tangent lines on I. [See Figure 3.23(a).]

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Figure 3.23(a)

1. Let f be differentiable on an open interval I. If the graph

of f is concave upward on I, then the graph of f lies

above all of its tangent lines on I. [See Figure 3.23(a).]

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Figure 3.23(a)

7.
2. Let f be differentiable on an open interval I. If the graph

of f is concave downward on I, then the graph of f lies

below all of its tangent lines on I. [See Figure 3.23(b).]

Figure 3.23(b) 7

of f is concave downward on I, then the graph of f lies

below all of its tangent lines on I. [See Figure 3.23(b).]

Figure 3.23(b) 7

8.
Concavity

To find the open intervals on which the

graph of a function f is concave upward

or concave downward, you need to find

the intervals on which f' is increasing or

For instance, the graph of

is concave downward on the open interval

because

is decreasing there. (See Figure 3.24)

Figure 3.24

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To find the open intervals on which the

graph of a function f is concave upward

or concave downward, you need to find

the intervals on which f' is increasing or

For instance, the graph of

is concave downward on the open interval

because

is decreasing there. (See Figure 3.24)

Figure 3.24

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9.
Concavity

Similarly, the graph of f is concave upward on the interval

because f' is increasing on .

The next theorem shows how to use the second derivative of

a function f to determine intervals on which the graph of f is

concave upward or concave downward.

To apply Theorem 3.7, locate the x-values at which

f" (x) = 0 or f" does not exist. Use these x-values to determine

test intervals. Finally, test the sign of f" (x) in each of the test

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Similarly, the graph of f is concave upward on the interval

because f' is increasing on .

The next theorem shows how to use the second derivative of

a function f to determine intervals on which the graph of f is

concave upward or concave downward.

To apply Theorem 3.7, locate the x-values at which

f" (x) = 0 or f" does not exist. Use these x-values to determine

test intervals. Finally, test the sign of f" (x) in each of the test

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10.
Example 1 – Determining Concavity

Determine the open intervals on which the graph of

is concave upward or downward.

Begin by observing that f is continuous on the entire real

Next, find the second derivative of f.

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Determine the open intervals on which the graph of

is concave upward or downward.

Begin by observing that f is continuous on the entire real

Next, find the second derivative of f.

10

11.
Example 1 – Solution cont’d

Because f ''(x) = 0 when x = ±1 and f'' is defined on the

entire line, you should test f'' in the intervals ,

(1, –1), and 11

Because f ''(x) = 0 when x = ±1 and f'' is defined on the

entire line, you should test f'' in the intervals ,

(1, –1), and 11

12.
Example 1 – Solution cont’d

The results are shown in the table and in Figure 3.25.

12

Figure 3.25

The results are shown in the table and in Figure 3.25.

12

Figure 3.25

13.
Concavity cont’d

The function given in Example 1 is continuous on the entire

real number line.

When there are x-values at which the function is not

continuous, these values should be used, along with the

points at which f"(x)= 0 or f"(x) does not exist, to form the

test intervals.

13

The function given in Example 1 is continuous on the entire

real number line.

When there are x-values at which the function is not

continuous, these values should be used, along with the

points at which f"(x)= 0 or f"(x) does not exist, to form the

test intervals.

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14.
Points of Inflection

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15.
Points of Inflection

If the tangent line to the graph exists at such a point where

the concavity changes, that point is a point of inflection.

Three types of points of inflection are shown in Figure 3.27.

The concavity of f changes at a point of inflection. Note that the graph crosses its tangent line at a

point of inflection.

Figure 3.27 15

If the tangent line to the graph exists at such a point where

the concavity changes, that point is a point of inflection.

Three types of points of inflection are shown in Figure 3.27.

The concavity of f changes at a point of inflection. Note that the graph crosses its tangent line at a

point of inflection.

Figure 3.27 15

16.
Points of Inflection

To locate possible points of inflection, you can determine

the values of x for which f"(x)= 0 or f"(x) does not exist. This

is similar to the procedure for locating relative extrema of f.

16

To locate possible points of inflection, you can determine

the values of x for which f"(x)= 0 or f"(x) does not exist. This

is similar to the procedure for locating relative extrema of f.

16

17.
Example 3 – Finding Points of Inflection

Determine the points of inflection and discuss the concavity

of the graph of

Differentiating twice produces the following.

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Determine the points of inflection and discuss the concavity

of the graph of

Differentiating twice produces the following.

17

18.
Example 3 – Solution cont’d

Setting f"(x) = 0, you can determine that the possible points

of inflection occur at x = 0 and x = 2.

By testing the intervals determined by these x-values, you

can conclude that they both yield points of inflection.

A summary of this testing is shown in

the table, and the graph of f is shown

in Figure 3.28.

Figure 3.28 18

Setting f"(x) = 0, you can determine that the possible points

of inflection occur at x = 0 and x = 2.

By testing the intervals determined by these x-values, you

can conclude that they both yield points of inflection.

A summary of this testing is shown in

the table, and the graph of f is shown

in Figure 3.28.

Figure 3.28 18

19.
Points of Inflection

The converse of Theorem 3.8 is not generally true. That is,

it is possible for the second derivative to be 0 at a point that

is not a point of inflection.

For instance, the graph of f(x) = x4 is shown in Figure 3.29.

The second derivative is 0 when

x= 0, but the point (0,0) is not a

point of inflection because the

graph of f is concave upward in

both intervals

Figure 3.29 19

The converse of Theorem 3.8 is not generally true. That is,

it is possible for the second derivative to be 0 at a point that

is not a point of inflection.

For instance, the graph of f(x) = x4 is shown in Figure 3.29.

The second derivative is 0 when

x= 0, but the point (0,0) is not a

point of inflection because the

graph of f is concave upward in

both intervals

Figure 3.29 19

20.
The Second Derivative Test

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21.
The Second Derivative Test

In addition to testing for concavity, the second derivative can

be used to perform a simple test for relative maxima and

The test is based on the fact that if the graph of f is concave

upward on an open interval containing c , and f' (c)=0, then

f(c) must be a relative minimum of f.

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In addition to testing for concavity, the second derivative can

be used to perform a simple test for relative maxima and

The test is based on the fact that if the graph of f is concave

upward on an open interval containing c , and f' (c)=0, then

f(c) must be a relative minimum of f.

21

22.
The Second Derivative Test

Similarly, if the graph of a function f is concave downward on

an open interval containing c, and f' (c)=0, then f(c) must be a

relative maximum of f (see Figure 3.30).

Figure 3.30

22

Similarly, if the graph of a function f is concave downward on

an open interval containing c, and f' (c)=0, then f(c) must be a

relative maximum of f (see Figure 3.30).

Figure 3.30

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23.
The Second Derivative Test

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24.
Example 4 – Using the Second Derivative Test

Find the extrema of f(x) = –3x5 + 5x3

Begin by finding the critical numbers of f.

From this derivative, you can see that x = –1, 0, and 1 are

the only critical numbers of f.

By finding the second derivative

you can apply the Second Derivative Test.

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Find the extrema of f(x) = –3x5 + 5x3

Begin by finding the critical numbers of f.

From this derivative, you can see that x = –1, 0, and 1 are

the only critical numbers of f.

By finding the second derivative

you can apply the Second Derivative Test.

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25.
Example 4 – Solution cont’d

Because the Second Derivative Test fails at (0, 0), you can

use the First Derivative Test and observe that f increases

to the left and right of x = 0.

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Because the Second Derivative Test fails at (0, 0), you can

use the First Derivative Test and observe that f increases

to the left and right of x = 0.

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26.
Example 4 – Solution cont’d

So, (0, 0) is neither a relative minimum nor a relative

maximum (even though the graph has a horizontal tangent

line at this point). The graph of f is shown in Figure 3.31.

Figure 3.31 26

So, (0, 0) is neither a relative minimum nor a relative

maximum (even though the graph has a horizontal tangent

line at this point). The graph of f is shown in Figure 3.31.

Figure 3.31 26