Calculus 2nd Edition Rogawski

SECTION 4.2 Extreme Values 191
20. Compute the critical points of h(t) = (t 2 − 1) 1/3 . Check that your
answer is consistent with Figure 17. Then find the extreme values of
h(t) on [0, 1] and [0, 2].
−2
−1
1
−1
h(t)
1 2
FIGURE 17 Graph of h(t) = (t 2 − 1) 1/3 .
21. Plot f (x) = 2 √ x − x on [0, 4] and determine the maximum
value graphically. Then verify your answer using calculus.
22. Plot f (x) = 2x 3 − 9x 2 + 12x on [0, 3] and locate the extreme
values graphically. Then verify your answer using calculus.
23. Approximate the critical points of g(x) = x cos x on
I =[0, 2π], and estimate the minimum value of g(x) on I.
24. Approximate the critical points of g(x) = tan 2 x − 5x on
I = ( − π 2 , π )
2 , and estimate the minimum value of g(x) on I.
In Exercises 25–50, find the min and max of the function on the given
interval by comparing values at the critical points and endpoints.
25. y = 2x 2 + 4x + 5, [−2, 2] 26. y = 2x 2 + 4x + 5, [0, 2]
27. y = 6t − t 2 , [0, 5] 28. y = 6t − t 2 , [4, 6]
29. y = x 3 − 6x 2 + 8, [1, 6]
30. y = x 3 + x 2 − x, [−2, 2] 31. y = 2t 3 + 3t 2 , [1, 2]
32. y = x 3 − 12x 2 + 21x, [0, 2] 33. y = z 5 − 80z, [−3, 3]
34. y = 2x 5 + 5x 2 , [−2, 2] 35. y = x2 + 1
, [5, 6]
x − 4
36. y = 1 − x
4x
x 2 , [1, 4] 37. y = x − , [0, 3]
+ 3x x + 1
38. y = 2 √ x 2 + 1 − x, [0, 2]
39. y = (2 + x) √ 2 + (2 − x) 2 , [0, 2]
40. y = √ 1 + x 2 − 2x, [0, 1]
41. y = √ x + x 2 − 2 √ x, [0, 4]
42. y = (t − t 2 ) 1/3 , [−1, 2] 43. y = sin x cos x,
44. y = x + sin x, [0, 2π] 45. y = √ 2 θ − sec θ,
t
[
0,
π
2
]
[
0,
π
3
]
46. y = cos θ + sin θ, [0, 2π] 47. y = θ − 2 sin θ, [0, 2π]
48. y = 4 sin 3 θ − 3 cos 2 θ, [0, 2π]
49. y = tan x − 2x, [0, 1]
50. y = sec 2 x − 2 tan x, [−π/6, π/3]
51. Let f(θ) = 2 sin 2θ + sin 4θ.
(a) Show that θ is a critical point if cos 4θ = − cos 2θ.
(b) Show, using a unit circle, that cos θ 1 = − cos θ 2 if and only if
θ 1 = π ± θ 2 + 2πk for an integer k.
(c) Show that cos 4θ = − cos 2θ if and only if θ = π 2 + πk or θ =
π
6 + ( π
)
3 k.
(d) Find the six critical points of f(θ) on [0, 2π] and find the extreme
values of f(θ) on this interval.
(e) Check your results against a graph of f(θ).
52. Find the critical points of f (x) = 2 cos 3x + 3 cos 2x in
[0, 2π]. Check your answer against a graph of f (x).
In Exercises 53–56, find the critical points and the extreme values on
[0, 4]. In Exercises 55 and 56, refer to Figure 18.
53. y =|x − 2| 54. y =|3x − 9|
55. y =|x 2 + 4x − 12| 56. y =|cos x|
30
20
10
y
−6
2
y = |x 2 + 4x − 12|
x
−
π
2
FIGURE 18
1
y
y = |cos x|
In Exercises 57–60, verify Rolle’s Theorem for the given interval.
57. f (x) = x + x −1 [
, 12
, 2 ] [
58. f (x) = sin x, π4
, 3π ]
4
59. f (x) = x2 , [3, 5]
8x − 15
60. f (x) = sin 2 x − cos 2 x,
[ π4
, 3π ]
4
61. Prove that f (x) = x 5 + 2x 3 + 4x − 12 has precisely one real root.
62. Prove that f (x) = x 3 + 3x 2 + 6x has precisely one real root.
63. Prove that f (x) = x 4 + 5x 3 + 4x has no root c satisfying c>0.
Hint: Note that x = 0 is a root and apply Rolle’s Theorem.
64. Prove that c = 4 is the largest root of f (x) = x 4 − 8x 2 − 128.
65. The position of a mass oscillating at the end of a spring is s(t) =
A sin ωt, where A is the amplitude and ω is the angular frequency.
Show that the speed |v(t)| is at a maximum when the acceleration a(t)
is zero and that |a(t)| is at a maximum when v(t) is zero.
66. The concentration C(t) (in mg/cm 3 ) of a drug in a patient’s bloodstream
after t hours is
C(t) = 0.016t
t 2 + 4t + 4
Find the maximum concentration in the time interval [0, 8] and the time
at which it occurs.
π
2
π
3π
2
x