116 0 CHAPTER 15
Fig. 15-46
15.57
Sketch the graph of /'(*), if Fig- 15-46 gives the graph of f(x).
Fig. 15-47
I A possible graph for /'(*) is shown in Fig. 15-47. At x = -2, /'(*) = 0. To the left of x=-2, f'(x)
is negative and approaches -°° as x-*-<*>. From x = -2 to x = -1, /'(x) is positive and increasing.
At x = -1, the graph of/has an inflection point, where /'(*) reaches a relative maximum. From x = -1
to x = 0, /'(*) is positive and decreasing to 0. From x = 0 to x-l, /'(*) is negative and decreasing.
At je = l, where the graph of /has an inflection point, the graph of/' has a relative minimum. From x = l
to ^ = 2, /'(*) is negative and increasing, reaching 0 at x = 2. From x = 2 to x = 3, /'(*) is positive
and increasing toward +°°. At * = 3, /is not differentiable, so/'is not defined. The graph of/'has * = 3
as a vertical asymptote. For x>3, /'(x) has a constant negative value (approximately -1).
Problems 15.58 to 15.61 refer to the function f(x) graphed in Fig. 15-48.
Fig. 15-48
Fig. 15-49
15.58 Which of the functions /(*)-!, /(*-!), /(-*), or /'(*) is graphed in Fig. 15-49?
I This is the graph of /(x - 1). It is obtained by shifting the graph of f(x) one unit to the right.
15.59 Which of the functions /(*)-!, /(*-!), /(-*), or f'(x) is graphed in Fig. 15-50?
I This is the graph of /(—Jc). It is obtained by reflecting the graph of /(*) in the y-axis.
Fig. 15-50