Increasing and Decreasing Functions and Relative Extrema

1. Chapter 5 Graphs and the Derivative

2. Section 5.1 Increasing and Decreasing Functions

3. Figure 2

5. Your Turn 1 Find where the function is increasing and decreasing. Solution: Moving from left to right, the function is decreasing for x-values up to −1, then increasing for x-values from to −1 to 2. For x-values from 2 to 4, decreasing and increasing for all x-values larger than 4. In interval notation, the function is increasing on (−1, 2) and (4, ∞). Function is decreasing on (−∞, −1) and (2,4).

6. Figure 4 - 5

10. Figure 6

11. Figure 7

12. Your Turn 2 Find the intervals in which is increasing or decreasing. Graph the function. Solution: Here To find the critical numbers, set this derivative equal to 0 and solve the resulting equation by factoring. Continued

13. Your Turn 2 Continued The tangent line is horizontal at −3 or 5/3. Since there are no values of x where derivative fails to exist, the only critical numbers are −3 and 5/3. To determine where the function is increasing or decreasing, locate −3 and 5/3 on a number line. The number line is divided into three intervals namely, (−∞, −3), ( −3, 5/3), and (5/3, ∞). Now choose any value of x in the interval (−∞, −3), choosing − 4 and evaluating which is negative. Therefore, f is decreasing on (−∞, −3). Continued

14. Your Turn 2 Continued Now choose any value of x in the interval ( −3, 5/3), choosing x = 0 and evaluating which is positive. Therefore, f is increasing on ( −3, 5/3). Now choose any value of x in the interval (5/3, ∞), choosing x = 2 and evaluating which is negative. Therefore, f is decreasing on (5/3, ∞). Continued

15. Your Turn 2 Continued

16. Figure 8

17. Figure 9

18. Figure 10

19. Figure 14

20. Section 5.2 Relative Extrema

22. Your Turn 1 Identify the x-values of all points where the graph has relative extrema. Solution:

23. Figure 16

24. Figure 17

26. Figure 18

29. Figure 19

30. Figure 20

31. Your Turn 2 Find all relative extrema Solution: Here To find the critical numbers, set this derivative equal to 0 and solve the resulting equation by factoring. Continued

32. Your Turn 2 Continued The tangent line is horizontal at −3 or 5/3. Since there are no values of x where derivative fails to exist, the only critical numbers are −3 and 5/3. The number line is divided into three intervals namely, (−∞, −3), ( −3, 5/3), and (5/3, ∞). Any number from each of the three intervals can be used as a test point to find the sign of in each interval. Using − 4, 0, and 2 gives the following information. Continued

33. Your Turn 2 Continued Thus derivative is negative on(−∞, −3), positive on ( −3, 5/3), and negative on (5/3, ∞). Using the first derivative test, this means that the function has a relative maximum of f (5/3) = 670/27 and f has a relative minimum of f (−3) = −26.

34. Figure 21

35. Figure 23 - 24

36. Figure 25 - 26

37. Figure 27

38. Section 5.3 Higher Derivatives, Concavity, and the Second Derivative Test

40. Your Turn 1 Solution: To find the second derivative of f (x), find the first derivative, and then take its derivative.

41. Your Turn 2(a) Solution : Here, using the chain rule,

42. Your Turn 2(b) Solution: Use the product rule,

43. Your Turn 2(c) Solution: Here, we need the quotient rule.

44. Figure 28

45. Figure 29

46. Figure 30

47. Figure 31

48. Figure 32

49. Figure 33