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

292 Chapter 4 Applications of Differentiation
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4. Draw the graph of a function that is continuous on f0, 8g
where f s0d − 1 and f s8d − 4 and that does not satisfy the
conclusion of the Mean Value Theorem on f0, 8g.
5–8 Verify that the function satisfies the three hypotheses of
Rolle’s Theorem on the given interval. Then find all numbers c
that satisfy the conclusion of Rolle’s Theorem.
5. f sxd − 2x 2 2 4x 1 5, f21, 3g
6. f sxd − x 3 2 2x 2 2 4x 1 2, f22, 2g
7. f sxd − sinsxy2d, fy2, 3y2g
8. f sxd − x 1 1yx, f 1 2 , 2g
9. Let f sxd − 1 2 x 2y3 . Show that f s21d − f s1d but there is
no number c in s21, 1d such that f 9scd − 0. Why does this
not contradict Rolle’s Theorem?
10. Let f sxd − tan x. Show that f s0d − f sd but there is no
number c in s0, d such that f 9scd − 0. Why does this not
contradict Rolle’s Theorem?
11–14 Verify that the function satisfies the hypotheses of the
Mean Value Theorem on the given interval. Then find all numbers
c that satisfy the conclusion of the Mean Value Theorem.
11. f sxd − 2x 2 2 3x 1 1, f0, 2g
12.
f sxd − x 3 2 3x 1 2, f22, 2g
13. f sxd − ln x, f1, 4g 14. f sxd − 1yx, f1, 3g
15–16 Find the number c that satisfies the conclusion of the
Mean Value Theorem on the given interval. Graph the function,
the secant line through the endpoints, and the tangent line
at sc, f scdd. Are the secant line and the tangent line parallel?
15. f sxd − sx , f0, 4g 16. f sxd − e 2x , f0, 2g
17. Let f sxd − sx 2 3d 22 . Show that there is no value of c
in s1, 4d such that f s4d 2 f s1d − f 9scds4 2 1d. Why does
this not contradict the Mean Value Theorem?
18. Let f sxd − 2 2 | 2x 2 1 | . Show that there is no value of
c such that f s3d 2 f s0d − f 9scds3 2 0d. Why does this not
contradict the Mean Value Theorem?
19–20 Show that the equation has exactly one real root.
19. 2x 1 cos x − 0 20. x 3 1 e x − 0
21. Show that the equation x 3 2 15x 1 c − 0 has at most one
root in the interval f22, 2g.
22. Show that the equation x 4 1 4x 1 c − 0 has at most two
real roots.
23. (a) Show that a polynomial of degree 3 has at most three
real roots.
(b) Show that a polynomial of degree n has at most n real
roots.
24. (a) Suppose that f is differentiable on R and has two roots.
Show that f 9 has at least one root.
(b) Suppose f is twice differentiable on R and has three
roots. Show that f 0 has at least one real root.
(c) Can you generalize parts (a) and (b)?
25. If f s1d − 10 and f 9sxd > 2 for 1 < x < 4, how small can
f s4d possibly be?
26. Suppose that 3 < f 9sxd < 5 for all values of x. Show that
18 < f s8d 2 f s2d < 30.
27. Does there exist a function f such that f s0d − 21, f s2d − 4,
and f 9sxd < 2 for all x?
28. Suppose that f and t are continuous on fa, bg and differenti
able on sa, bd. Suppose also that f sad − tsad and
f 9sxd , t9sxd for a , x , b. Prove that f sbd , tsbd. [Hint:
Apply the Mean Value Theorem to the function h − f 2 t.]
29. Show that sin x , x if 0 , x , 2.
30. Suppose f is an odd function and is differentiable everywhere.
Prove that for every positive number b, there exists
a number c in s2b, bd such that f 9scd − f sbdyb.
31. Use the Mean Value Theorem to prove the inequality
| sin a 2 sin b | < | a 2 b |
for all a and b
32. If f 9sxd − c (c a constant) for all x, use Corollary 7 to show
that f sxd − cx 1 d for some constant d.
33. Let f sxd − 1yx and
tsxd −
1
x
1 1 1 x
if x . 0
if x , 0
Show that f 9sxd − t9sxd for all x in their domains. Can we
conclude from Corollary 7 that f 2 t is constant?
34. Use the method of Example 6 to prove the identity
2 sin 21 x − cos 21 s1 2 2x 2 d x > 0
35. Prove the identity arcsin x 2 1
x 1 1 − 2 arctan sx 2 2 .
36. At 2:00 pm a car’s speedometer reads 30 miyh. At 2:10 pm it
reads 50 miyh. Show that at some time between 2:00 and
2:10 the acceleration is exactly 120 miyh 2 .
37. Two runners start a race at the same time and finish in a tie.
Prove that at some time during the race they have the same
speed. [Hint: Consider f std − tstd 2 hstd, where t and h are
the position functions of the two runners.]
38. A number a is called a fixed point of a function f if
f sad − a. Prove that if f 9sxd ± 1 for all real numbers x, then
f has at most one fixed point.
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