1. Copyright © Cengage Learning. All rights reserved. 3 Applications of the Derivative

2. Copyright © Cengage Learning. All rights reserved. Concavity and the Second-Derivative Test 3.3

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

4. 44  Find the points of diminishing returns of input- output models. Objectives

5. 55 Concavity

6. 66 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

7. 77 This property of curving upward or downward is defined formally as the concavity of the graph of the function. Concavity

8. 88 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

9. 99 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

10. 10 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

11. 11 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.

12. 12 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

13. 13 The results are shown in the table and in Figure 3.23. Concavity Figure 3.23

14. 14 Points of Inflection

15. 15 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

16. 16 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

17. 17 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

18. 18 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.

19. 19 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

20. 20 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

21. 21 The Second-Derivative Test

22. 22 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

23. 23 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

24. 24 The Second-Derivative Test

25. 25 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.

26. 26 Example 5 – Solution cont’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.

27. 27 Example 5 – Solution cont’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

28. 28 Extended Application: Diminishing Returns

29. 29 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

30. 30 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

31. 31 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

32. 32 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.

33. 33 Example 6 – Solution Begin by finding the first and second derivatives. The second derivative is zero only when x = 20.

34. 34 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