Exercises 25-36 refer to Fig. 23, which contains the graph of f\'(x). 41. By loo

Question

Exercises 25-36 refer to Fig. 23, which contains the graph of f'(x). 41. By loo the derivative of the function f(x) curves 3 (6, 2) 01 2 345 6 42. Match Obsery Figure 23 25. Explain why f(x) must be increasing at x 6. 26. Explain why f(x) must be decreasing at x- 4. 27. Explain why f(x) has a relative maximum at x-3 28. Explain why f(x) has a relative minimum at x = 5. 29. Explain why f(x) must be concave up at x = 0. 30. Explain why f(x) must be concave down at x-2 31. Explain why f(x) has an inflection point at x 1. 32. Explain why f(x) has an inflection point at x = 4. 33. If f(6-3, what is the equation of the tangent line to the (b) 1 (c) T (d) l (e) T Conclu (B) V (C) V graph of y-f(x) at x 6? 34. If f(6-8, what is an approximate value of f(65)? 35. If f(0) 3, what is an approximate value of f(.25)? 36. If (0) 3, what is the equation of the tangent line to the (D) V (E) j 43. Numbe United functic are sho graph of y-f(x) atx-0? 37. Level of Water from Melting Snow Melting snow causes a river to overflow its banks. Let h(t) denote the number of inches of water on Main Street t hours after the melting begins. (a) If i'(100) , by approximately how much will the water level change during the next half hour? Which of the following two conditions is the better news? (0) h(100) 3, h'(100)2, h"(100)--5 (ii) h(100)3, h'(100)--2, h"(100) (b)

Explanation / Answer

Solution -

28 . From graph of f'(x) we can see f'(x) = 0 at x = 5 .

It means x = 5 is a critical point of f(x).

At x = 5 , we can see graph of f(x) is increasing

It means f''(5) >0

So there is a relative minimum at x = 5.

31)

We can see at x = 1

Derivative of f'(x) is 0 because slope of tangent is 0 .

It means f''(x) = 0 at x = 1.

Graph of f'(x) increasing at left of x= 1 and decreasing at right so it is point of

inflection for graph of f(x).

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