James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

Section 4.3 How Derivatives Affect the Shape of a Graph 301
8. The graph of the first derivative f 9 of a function f is shown.
(a) On what intervals is f increasing? Explain.
(b) At what values of x does f have a local maximum or
minimum? Explain.
(c) On what intervals is f concave upward or concave downward?
Explain.
(d) What are the x-coordinates of the inflection points of f ?
Why?
y
0
y=fª(x)
2 4 6 8
9–18
(a) Find the intervals on which f is increasing or decreasing.
(b) Find the local maximum and minimum values of f.
(c) Find the intervals of concavity and the inflection points.
9. f sxd − x 3 2 3x 2 2 9x 1 4
10. f sxd − 2x 3 2 9x 2 1 12x 2 3
11. f sxd − x 4 2 2x 2 1 3 12. f sxd − x
x 2 1 1
13. f sxd − sin x 1 cos x, 0 < x < 2
14.
f sxd − cos 2 x 2 2 sin x, 0 < x < 2
15. f sxd − e 2x 1 e 2x 16. f sxd − x 2 ln x
17. f sxd − x 2 2 x 2 ln x
x
18. f sxd − x 4 e 2x
19–21 Find the local maximum and minimum values of f using
both the First and Second Derivative Tests. Which method do you
prefer?
19. f sxd − 1 1 3x 2 2 2x 3 20. f sxd − x 2
x 2 1
21. f sxd − sx 2 s 4 x
22. (a) Find the critical numbers of f sxd − x 4 sx 2 1d 3 .
(b) What does the Second Derivative Test tell you about the
behavior of f at these critical numbers?
(c) What does the First Derivative Test tell you?
23. Suppose f 0 is continuous on s2`, `d.
(a) If f 9s2d − 0 and f 0s2d − 25, what can you say about f ?
(b) If f 9s6d − 0 and f 0s6d − 0, what can you say about f ?
24–31 Sketch the graph of a function that satisfies all of the given
conditions.
24. (a) f9sxd , 0 and f 0sxd , 0 for all x
(b) f9sxd . 0 and f 0sxd . 0 for all x
25. (a) f9sxd . 0 and f 0sxd , 0 for all x
(b) f9sxd , 0 and f 0sxd . 0 for all x
26. Vertical asymptote x − 0, f 9sxd . 0 if x , 22,
f 9sxd , 0 if x . 22 sx ± 0d,
f 0sxd , 0 if x , 0, f 0sxd . 0 if x . 0
27. f 9s0d − f 9s2d − f 9s4d − 0,
f 9sxd . 0 if x , 0 or 2 , x , 4,
f 9sxd , 0 if 0 , x , 2 or x . 4,
f 0sxd . 0 if 1 , x , 3, f 0sxd , 0 if x , 1 or x . 3
28. f 9sxd . 0 for all x ± 1, vertical asymptote x − 1,
f 99sxd . 0 if x , 1 or x . 3, f 99sxd , 0 if 1 , x , 3
29. f 9s5d − 0, f 9sxd , 0 when x , 5,
f 9sxd . 0 when x . 5, f 99s2d − 0, f 99s8d − 0,
f 99sxd , 0 when x , 2 or x . 8,
f 99sxd . 0 for 2 , x , 8, lim
fsxd − 3,
xl`
30. f 9s0d − f 9s4d − 0, f 9sxd − 1 if x , 21,
f 9sxd . 0 if 0 , x , 2,
f 9sxd , 0 if 21 , x , 0 or 2 , x , 4 or x . 4,
lim f9sxd − `, lim f9sxd − 2`,
x l22 x l21 f 99sxd . 0 if 21 , x , 2 or 2 , x , 4,
f 0sxd , 0 if x . 4
31. f 9sxd . 0 if x ± 2, f 99sxd . 0 if x , 2,
f 99sxd , 0 if x . 2, f has inflection point s2, 5d,
lim f sxd − 8, lim f sxd − 0
xl` xl 2`
lim f sxd − 3
xl 2`
32. Suppose fs3d − 2, f 9s3d − 1 2 , and f 9sxd . 0 and f 0sxd , 0
for all x.
(a) Sketch a possible graph for f.
(b) How many solutions does the equation f sxd − 0 have?
Why?
(c) Is it possible that f 9s2d − 1 3 ? Why?
33. Suppose f is a continuous function where fsxd . 0 for all x,
f s0d − 4, f 9sxd . 0 if x , 0 or x . 2, f 9sxd , 0
if 0 , x , 2, f 99s21d − f 99s1d − 0, f 99sxd . 0 if
x , 21 or x . 1, f 99sxd , 0 if 21 , x , 1.
(a) Can f have an absolute maximum? If so, sketch a possible
graph of f. If not, explain why.
(b) Can f have an absolute minimum? If so, sketch a possible
graph of f. If not, explain why.
(c) Sketch a possible graph for f that does not achieve an
absolute minimum.
34. The graph of a function y − f sxd is shown. At which point(s)
are the following true?
(a) dy
dx and d 2 y
are both positive.
2
dx
(b) dy
dx and d 2 y
2
are both negative.
dx
(c) dy
dx is negative but d 2 y
is positive.
2
dx
y
0
A
B
C
D
E
x
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