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3 Applicationsof the Derivative

Copyright © Cengage Learning. All rights reserved.

Concavity and the Second-Derivative Test

3.3

33

Determine the intervals on which the graphs of functions are concave upward or concave downward.

Find the points of inflection of the graphs of functions.

Use the Second-Derivative Test to find the relative extrema of functions.

Objectives

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Find the points of diminishing returns of input-output models.

Objectives

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Concavity

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You already know that locating the intervals over which a function f increases or decreases helps to describe its graph.

In this section, you will see that locating the intervals on which f increases or decreases can determine where the graph of f is curving upward or curving downward.

Concavity

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This property of curving upward or downward is defined formally as the concavity of the graph of the function.

Concavity

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In Figure 3.20, you can observe the following graphical interpretation of concavity.

1. A curve that is concave upward lies above its tangent line.

2. A curve that is concave downward lies below its tangent line.

Concavity

Figure 3.20

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To find the open intervals on which the graph of a function is concave upward or concave downward, you can use the second derivative of the function as follows.

Concavity

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For a continuous function f, you can find the open intervals on which the graph of f is concave upward and concave downward as follows.

Concavity

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Example 2 – Applying the Test for Concavity

Determine the open intervals on which the graph of

is concave upward or concave downward.

Solution:

Begin by finding the second derivative of f.

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Example 2 – Solution

From this, you can see that f (x) is defined for all real numbers and f (x) = 0 when x = 1. So, you can test the concavity of f by testing the intervals

cont’d

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The results are shown in the table and in Figure 3.23.

Concavity

Figure 3.23

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Points of Inflection

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If the tangent line to a graph exists at a point at which the concavity changes, then the point is a point of inflection. Three examples of inflection points are shown in Figure 3.24. (Note that the third graph has a vertical tangent line at its point of inflection.)

Points of Inflection

The graph crosses its tangent line at a point of inflection.

Figure 3.24

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Because a point of inflection occurs where the concavity of a graph changes, it must be true that at such points the sign of f changes.

Points of Inflection

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So, to locate possible points of inflection, you need to determine only the values of x for which f (x) = 0 or for which f (x) does not exist.

This parallels the procedure for locating the relative extrema of f by determining the critical numbers of f.

Points of Inflection

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Example 4 – Finding Points of Inflection

Discuss the concavity of the graph of

and find its points of inflection.

Solution:

Begin by finding the second derivative of f.

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Example 4 – Solution

From this, you can see that the possible points of inflection occur at and

After testing the intervals and you can determine that the graph is concave upward on concave downward on and concave upward on

cont’d

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Example 4 – Solution

Because the concavity changes at

and you can

conclude that the graph of f has

points of inflection at these

x-values, as shown in Figure 3.26.

The points of inflection are

cont’d

Two Points of Inflection

Figure 3.26

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The Second-Derivative Test

2222

The second derivative can be used to perform a simple test for relative minima and relative maxima.

If f is a function such that f (c) = 0 and the graph of f is concave upward at x = c, then f (c) is a relative minimum of f.

The Second-Derivative Test

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Similarly, if f is a function such that f (c) = 0 and the graph of f is concave downward at x = c, then f (c) is a relative maximum of f, as shown in Figure 3.28.

The Second-Derivative Test

Figure 3.28

Relative maximum Relative minimum

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The Second-Derivative Test

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Example 5 – Using the Second-Derivative Test

Find the relative extrema of

Solution:

Begin by finding the first derivative of f.

From this derivative, you can see that x = 0, x = –1, and x = 1 are the only critical numbers of f.

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Example 5 – Solutioncont’d

Using the second derivative

you can apply the Second-Derivative Test, as shown.

Because the Second-Derivative Test fails at (0, 0) you can use the First-Derivative Test and observe that f is positive on both sides of x = 0. So, (0, 0) is neither a relative minimum nor a relative maximum.

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Example 5 – Solutioncont’d

A test for concavity would show that (0, 0) is a point of inflection. The graph of f is shown in Figure 3.29.

Figure 3.29

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Extended Application: Diminishing Returns

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In economics, the notion of concavity is related to the concept of diminishing returns. Consider a function

where x measures input (in dollars) and y measures output (in dollars).

Extended Application: Diminishing Returns

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In Figure 3.30, notice that the graph of this function is concave upward on the interval (a, c) and is concave downward on the interval (c, b).

Extended Application: Diminishing Returns

Figure 3.30

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On the interval (a, c), each additional dollar of input returns more than the previous input dollar.

By contrast, on the interval (c, b) each additional dollar of input returns less than the previous input dollar.

The point (c, f (c)) is called the point of diminishing returns.

An increased investment beyond this point is usually considered a poor use of capital.

Extended Application: Diminishing Returns

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Example 6 – Exploring Diminishing Returns

By increasing its advertising cost x (in thousands of dollars) for a product, a company discovers that it can increase the sales y (in thousands of dollars) according to the model

Find the point of diminishing returns for this product.

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Example 6 – Solution

Begin by finding the first and second derivatives.

The second derivative is zero only when x = 20.

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Example 6 – Solution

By testing for concavity on the

intervals (0, 20) and (20, 40),

you can conclude that the graph

has a point of diminishing returns

when x = 20, as shown in

Figure 3.31.

So, the point of diminishing

returns for this product occurs

when $20,000 is spent on

advertising.

cont’d

Figure 3.31