118 SECTION 4.1 CHAPTER 4
52. A proof is outlined just below the statement of the theorem
53. f(x) = 2 sec x(sec x tan x) = 2 sec2x tan x and g(x) = 2 tan x sec2x.
Therefore, f(x) = g(x) for all x∈ I.
54. Evaluating sec2x− tan2x = C at x = 0 gives C = 1.
55. Let f and g be functions such that f(x) =−g(x) and g(x) = f (x). Then:
(a) Differentiating f2(x) + g2(x) with respect to x, we have
2f (x)f(x) + 2g(x)g(x) =−2f(x)g(x) + 2g(x)f(x) = 0.
Thus, f2(x) + g2(x) = C (constant).
(b) f (0) = 0 and g(0) = 1 implies C = 1.
(c) The functions f (x) = sin x, g(x) = cos x have these properties.
56. (a) Let h(x) = f (x)− g(x). Then h(x) = f(x)− g(x) > 0 on (0, c), and h is increasing on (0, c).
Since h(0) = f (0)− g(0) = 0, it follows that h(x) > 0 on (0, c). Thus, f(x) > g(x) on (0, c).
(b) Again let h(x) = f (x)− g(x). Then h is increasing on (−c, 0) which implies that h(x) < 0 on this interval since h(0) = 0. Therefore, f (x) < g(x) on (−c, 0).
57. Let f (x) = tan x and g(x) = x for x∈ [0, π/2). Then f(0) = g(0) = 0 and f(x) = sec2x > g(x) = 1 for x∈ (0, π/2). Thus, tan x > x for x ∈ (0, π/2) by Exercise 56(a).
58. Let f (x) = cos x−
1−12x2
for x∈ [0, ∞). Then f(0) = 0 and f(x) =− sin x + x = x − sin x > 0 for x∈ (0, ∞) by Exercise 51 (b). Thus, f(x) > 0 for x ∈ (0, ∞) which implies cos x > 1 −12x2 on (0,∞).
59. Choose an integer n > 1. Let f (x) = (1 + x)n and g(x) = 1 + nx, x > 0. Then, f (0) = g(0) = 1 and f(x) = n(1 + x)n−1> g(x) = n since (1 + x)n−1> 1 for x > 0. The result follows from Exercise 56(a).
60. Let f (x) = sin x−
x−16x3
. Then f (0) = 0 and f(x) = cos x−
1−12x2
> 0 by Exercise 58.
Therefore, f (x) > f (0) = 0 for all x∈ (0, ∞) which implies sin x > x − 16x3 on (0,∞).
61. 4◦∼= 0.06981 radians. By Exercises 51 and 60, 0.6981−(0.6981)3
6 = 0.06975 < sin 4◦< 0.6981
SECTION 4.2 131
62. (a) Let f (x) = cos x− (1 − 1 2x2+ 1
24x4). Then f (0) = 0 and f(x) =− sin x + x −x3
6 < 0 by Exercise 60. Therefore, f (x) < f (0) = 0 on all x∈ (0, ∞), which implies cos x < 1 −1
2x2+ 1 24x4 on (0,∞).
(b) 6◦ = π
30. Using this for x in 1−1
2x2< cos x < 1−1 2x2+ 1
24x4,
=⇒ 0.994517 < cos 6◦< 0.994522.
63. Let f (x) = 3x4− 10x3− 4x2+ 10x + 9, x∈ [−2, 5]. Then f(x) = 12x3− 30x2− 8x + 10.
f(x) = 0 at x ∼=−0.633, 0.5, 2.633 f is decreasing on [−2, −0.633]
and [0.5, 2.633]
f is increasing on [−0.633, 0.5]
and [2.633, 5]
64. Let f (x) = 2x3− x2− 13x − 6, x ∈ [−3, 4]. Then f(x) = 6x2− 2x − 13.
f(x) = 0 at x ∼=−1.315, 1.648 f is decreasing on [−1.315, 1.648]
f is increasing on [−3, −1.315] and [1.648, 4]
65. Let f (x) = x cos x− 3 sin 2x, x ∈ [0, 6]. Then f(x) = cos x− x sin x − 6 cos 2x.
f(x) = 0 at x ∼= 0.770, 2.155, 3.798, 5.812 f is decreasing on [0, 0.770], [2.155, 3.798]
and [5.812, 6]
f is increasing on [0.770, 2.155]
and [3.798, 5.812]
66. Let f (x) = x4+ 3x3− 2x2+ 4x + 4, x∈ [−5, 3]. Then f(x) = 4x3+ 9x2− 4x + 4.
f(x) = 0 at x ∼=−2.747 f is decreasing on [−5, −2.747]
f is increasing on [−2.747, 3]
132 SECTION 4.3
67. (a) f(x) = 0 at x = 0, π2, π, 3π2 , 2π (b) f(x) > 0 on π,3π2
∪3π
2, 2π (c) f(x) < 0 on
0,π2
∪π
2, π 68. (b) f(x) > 0 on (−∞, ∞)
69. (a) f(x) = 0 at x = 0 (b) f(x) > 0 on (0,∞) (c) f(x) < 0 on (−∞, 0)
70. (a) f(x) = 0 at x =−12, 85, 3 (b) f(x) > 0 on
−∞, −12
∪
−12,85
∪ (3, ∞) (c) f(x) < 0 on 8
5, 3 71. f = C, constant; f(x)≡ 0
SECTION 4.3
1. f(x) = 3x2+ 3 > 0; no critical pts, no local extreme values 2. f(x) = 8x3− 8x = 8x(x2− 1); critical pts−1, 0, 1
f(x) = 24x2− 8; f(−1) = f(1) = 16 > 0, f(0) =−8 < 0;
f (0) = 6 local max, f (−1) = 4 local min, f(1) = 4 local min
3. f(x) = 1− 1
x2; critical pts−1, 1 f(x) = 2
x3, f(−1) = −2, f(1) = 2 f (−1) = −2 local max, f(1) = 2 local min 4. f(x) = 2x + 6
x3 =2x4+ 6
x3 ; no critical pts (note: 0 is not in the domain of f ), no local extreme values
5. f(x) = 2x− 3x2= x(2− 3x); critical pts 0,23 f(x) = 2− 6x; f(0) = 2, f(23) =−2 f (0) = 0 local min, f (23) = 274 local max
6. f(x) =−2(1 − x)(1 + x) + (1 − x)2= (x− 1)(3x + 1); critical pts −13, 1 f(x) = (1 + 3x) + 3(x− 1) = 2(3x − 1); f
−13
=−4, f(1) = 4 f
−13
=3227 local max, f (1) = 0 local min
7. f(x) = 2
(1− x)2; no critical pts, no local extreme values
8. f(x) = (2 + x)(−3) − (2 − 3x)(1)
(2 + x)2 =− 8
(2 + x)2; no critical pts (note:−2 is not in the domain of f ), no local extreme values
SECTION 4.3 133
9. f(x) =−2(2x + 1)
x2(x + 1)2; critical pt−1 2
f
−12
=−8 local max
10. f (x) =
⎧⎪
⎪⎨
⎪⎪
⎩
x2− 16, x <−4 16− x2, −4 ≤ x < 4 x2− 16, x≥ 4
f(x) =
⎧⎪
⎪⎨
⎪⎪
⎩
2x, x <−4
−2x, −4 < x < 4 2x, x > 4
critical pts−4, 0, 4; f (−4) = f(4) = 0 local minima, f(0) = 16 local max 11. f(x) = x2(5x− 3)(x − 1); critical pts 0, 35, 1
f
3 5
= 2233
55 local max f (1) = 0 local min no local extreme at 0
12. f(x) = 3
x− 2 x + 2
2
4
(x + 2)2 ≥ 0; critical pt 2, no local extreme values 13. f(x) = (5− 8x)(x − 1)2; critical pts 58, 1
f5
8
=204827 local max no local extreme at 1 14. f(x) =−(1 + x)3+ (1− x)(3)(1 + x)2= 2(1 + x)2(1− 2x); critical pts −1,12
f1
2
= 2716 local max no local extreme at −1 15. f(x) =x(2 + x)
(1 + x)2; critical pts−2, 0
f (−2) = −4 local max f (0) = 0 local min 16. f(x) = (1− x)1/3−13x(1− x)−2/3= 3− 4x
3(1− x)2/3; critical pts 34, 1
f (3/4) = 44/33 local max no local extreme at 1 17. f(x) =13x(7x + 12)(x + 2)−2/3; critical pts−2, −127, 0
f
−127
= 14449 2
7
1/3
local max f (0) = 0 local min
134 SECTION 4.3
18. f(x) = −1
(x + 1)2 + 1
(x− 2)2 = 3(2x− 1)
(x + 1)2(x− 2)2; critical pt 12
f1
2
=43 local min
19. f (x) =
⎧⎪
⎪⎨
⎪⎪
⎩
2− 3x, x≤ −12 x + 4, −12< x < 3 3x− 2, 3≤ x
f(x) =
⎧⎪
⎪⎨
⎪⎪
⎩
−3, x <−12 1, −12 < x < 3 3, 3 < x critical pts−12, 3
f
−12
= 72 local min no local extreme at 3
20. f(x) = 73x4/3−73x−2/3=7 3
x2− 1
x2/3 ; critical pts−1, 0, 1
f (−1) = 6 local max, f (1) =−6 local min no local extreme at 0 21. f(x) = 23x−4/3(x− 1); critical pt 1
f (1) = 3 local min no local extreme at 0
22. f(x) = (x + 1)3x2− x3
(x + 1)2 = x2(2x + 3)
(x + 1)2 ; critical pts−32, 0
f
−32
= 274 local min no local extreme at 0
23. f(x) = cos x− sin x; critical pts 14π, 54π f(x) =− sin x − cos x, f1
4π
=−√
2, f5
4π
=√ 2 f (14π) =√
2 local max, f (54π) =−√
2 local min 24. f(x) = 1− 2 sin 2x; critical pts 12π, 5π12
f (121π) = π 12+
√3
2 local max f (5
12π) = 5π 12 −
√3
2 local min
SECTION 4.3 135
25. f(x) = cos x (2 sin x−√
3 ); critical pts 13π, 12π, 23π
f (13π) = f (23π) =−34 local mins f (12π) = 1−√
3 local max
26. f(x) = 2 sin x cos x; critical pts 12π, π, 32π
f1
2π
= 1 = f3
2π
local max f (π) = 0 local min
27. f(x) = cos2x− sin2x− 3 cos x + 2 = (2 cos x − 1)(cos x − 1) critical pts 13π, 53π f1
3π
=23π−54√
3 local min f5
3π
=103π +54√
3 local max
28. f(x) = 6 sin2x cos x− 3 cos x = 3 cos x(2 sin2x− 1); critical pts 14π, 12π, 34π
f (14π) = f (34π) =−√
2 local mins f (12π) =−1 local max
29. (a) f increases on [−2, 0] and [3, ∞); f decreases on (−∞, −2] and [0, 3].
(b) f (−2) and f(3) are local minima; f(0) = 1 is a local maximum.
30. (a) f increases on (−∞, −1] and [0, ∞); f decreases on [−1, 0].
(b) f (−1) is a local maximum; f(0) = 1 is a local minimum.
31. Let h(x) = f (x)− g(x). Then h(x) gives the vertical separation between graphs of f and g at x. If h has a maximum at c, then h(c) = 0. Since h(x) = f(x)− g(x), h(c) = 0 implies f(c) = g(c).
Thus the lines tangent to the graphs of f and g are parallel at x = c.
32. Set g(x) =−f(−x) and apply the proof of the second derivative test already given.
33. Solving f(x) = 2ax + b = 0 gives a critical point at x =− b
2a. Since f(x) = 2a, f has a local maximum at− b
2a if a < 0 and a local minimum at− b
2a if a > 0.
34. Setting f(x) = 3ax2+ 2bx + c = 0 and checking the discriminant, we get (1) 2 local extrema if b2> 3ac
(2) 1 local extrema if b2= 3ac (3) 0 local extrema if b2< 3ac
136 SECTION 4.3
35. P (x) = x4− 8x3+ 22x2− 24x + 4
P(x) = 4x3− 24x2+ 44x− 24 P(x) = 12x2− 48x + 44
Since P(1) = 0, P(x) is divisible by x− 1. Division by x − 1 gives P(x) = (x− 1)
4x2− 20x + 24
= 4(x− 1)(x − 2)(x − 3).
The critical pts are 1, 2, 3. Since
P(1) > 0, P(2) < 0, P(3) > 0,
P (1) =−5 is a local min, P (2) = −4 is a local max, and P (3) = −5 is a local min.
Since P(x) < 0 for x < 0, P decreases on (−∞, 0]. Since P (0) > 0, P does not take on the value 0 on (−∞, 0].
Since P (0) > 0 and P (1) < 0, P takes on the value 0 at least once on (0, 1). Since P(x) < 0 on (0, 1), P decreases on [0, 1]. It follows that P takes on the value zero only once on [0, 1].
Since P(x) > 0 on (1, 2) and P(x) < 0 on (2, 3), P increases on [1, 2] and decreases on [2, 3]. Since P (1), P (2), P (3) are all negative, P cannot take on the value 0 between 1 and 3.
Since P (3) < 0 and P (100) > 0, P takes on the value 0 at least once on (3, 100). Since P(x) > 0 on (3, 100), P increases on [3, 100]. It follows that P takes on the value zero only once on [3, 100].
Since P(x) > 0 on (100,∞), P increases on [100, ∞). Since P (100) > 0, P does not take on the value 0 on [100,∞).
36. f has a local maximum at x = 0; f has a local minimum at x =−1 and x = 2.
37.
38. Let f (x) = Ax2+ Bx + C. Then f(x) = 2Ax + B.
f (−1) = 3 =⇒ A − B + C = 3; f(3) = −1 =⇒ 9A + 3B + C = −1 Since f has a minimum at x = 2, f(2) = 4A + B = 0
Solving for A, B, C, we get A = 12, B =−2, C = 12.
SECTION 4.3 137
39. Let f (x) = ax
x2+ b2. Then f(x) = a
b2− x2 (b2+ x2)2. Now f(0) = a
b2 = 1⇒ a = b2 and f(x) = b2
b2− x2 (b2+ x2)2
f(−2) = b2 b2− 4
(b2+ 4)2 = 0 ⇒ b = ±2 Thus, a = 4 and b =±2.
40. (a) f (x) = xp(1− x)q, p, q≥ 2; f(x) = xp−1(1− x)q−1[p− (p + q)x]
f(x) = 0 =⇒ x = 0, x = 1, x = p p + q (b) p even, p− 1 odd:
f has a local min at x = 0 (c) q even, q− 1 odd:
f has a local min at x = 1
(d) f
p p + q
=− (p + q)
p p + q
p−1 q p + q
q−1
< 0 ⇒ f has a local max at x = p p + q. 41. If p is a polynomial of degree n, then p has degree n− 1. This implies that phas at most
n− 1 zeros, and it follows that p has at most n − 1 local extreme values.
42. The function D(x) =
x2+ [f (x)]2 gives the distance from the origin to the point (x, f (x)) on the graph of f. Since the graph of f does not pass through the origin,
D(x) = x + f (x)f(x) x2+ [f (x)]2
is defined for all x∈ dom (f). Suppose that D has a local extreme value at c. Then D(c) = c + f (c)f(c)
c2+ [f (c)]2 = 0 ⇒ c + f(c)f(c) = 0 and f(c) =− c f (c)
Suppose that c= 0. The slope of the line through (0, 0) and (c, f(c)) is given by m1=f (c)
c and the slope of the tangent line to the graph of f at x = c is given by m2= f(c) =− c
f (c). Since m1m2=−1, these two lines are perpendicular. If c = 0, then the tangent line to the graph of f is horizontal and the line through (0, 0) and (0, f (0)) is vertical.
43. If f (x) = x4− 7x2− 8x − 3, then f(x) = 4x3− 14x − 8 and f(x) = 12x2− 14. Since f(2) =
−4 < 0 and f(3) = 58 > 0, f has at least one zero in (2, 3). Since f(x) > 0 for x∈ (2, 3), f is increasing on this interval and so it has exactly one zero. Thus, f has exactly one critical point c in (2, 3).
138 SECTION 4.3
44. If f (x) = sin x + x2
2 − 2x, then f(x) = cos x + x− 2 and f(x) =− sin x + 1. Since f(2) =
−0.4161 < 0 and f(3) = 0.01 > 0, f has at least one zero in (2, 3). Since f(x) > 0 for x∈ (2, 3), f is increasing on this interval and so it has exactly one zero. Thus, f has exactly one critical point c in (2, 3).
45. f (x) = ax2+ b
cx2+ d and f(x) = 2(ad− bc)x
(cx2+ d)2 ; x = 0 is a critical number.
f(x) = 2(ad− bc)(cx2− 4cx + d)
(cx2+ d)3 ; f(0) = 2(ad− bc) d2
Therefore, ad− bc > 0 implies that f(0) is a local minimum; ad − bc < 0 implies that f(0) is a local maximum.
46. Let δ be any positive number and consider f on the interval (−δ, δ). Let n be a positive integer such that
0 < 1
π
2+ 2nπ < δ and 0 < 1
−π2 + 2nπ < δ.
Then
f
1
π 2 + 2nπ
> 0 and f
1
−π2 + 2nπ
< 0.
Thus f takes on both positive and negative values in every interval centered at 0 and it follows that f cannot have a local maximum or minimum at 0.
47. (a)
critical points: x1∼=−0.692, x2∼= 2.248
local extreme values: f (−0.692) ∼= 29.342, f (2.248) ∼=−8.766
(b) f is increasing on [−3, −0.692], and [2.248, 4]; f is decreasing on [−0.692, 2.248]
48. (a)
SECTION 4.3 139
critical points: x1∼=−2.085, x2∼=−1, x3∼= 0.207, x4∼= 1.096, x5= 1.544
local extreme values: f (−2.085) ∼=−6.255, f(−1) = 7, f(0.207) ∼= 0.621, f (1.096) ∼= 7.097, f (1.544) ∼= 4.635
(b) f is increasing on [−2.085, −1], [0.207, 1.096], and [1.544, 4]
f is decreasing on [−4, −2.085], [−1, 0.207], and [1.096, 1.544]
49. (a)
critical points: x1∼=−2.201, x2∼=−0.654, x3∼= 0.654, x4∼= 2.201
local extreme values: f (−2.204) ∼= 2.226, f (−0.654) ∼=−6.634, f(0.654) ∼= 6.634, f (2.204) ∼=−2.226
(b) f is increasing on [−3. − 2.204], [−0.654, 0.654], and [2.204, 3]
f is decreasing on [−2.204, −0.654], and [0.654, 2.204]
50. f(x) = 0 at x = 2, 3; f (2) = 0 is a local minimum.
51. f(x) > 0 on 2
3,∞
; f has no local extrema.
52. f(x) = 0 at the multiples of 14π; f (0) = 1 is a local maximum, f (π/4) = 0 is a local minimum, f (π/2) = 1 is a local maximum, and so on.
53.
critical points of f : x1∼=−1.326, x2= 0, x3∼= 1.816
f(−1.326) ∼=−4 < 0 ⇒ f has a local maximum at x = −1.326 f(0) = 4 > 0 ⇒ f has a local minimum at x = 0
f(1.816) ∼=−4 ⇒ f has a local maximum at x = 1.816
140 SECTION 4.4
54.
critical number of f : x1∼=−1.935
f(−1.935) ∼= 14.60 > 0 ⇒ f has a local minimum at x = −1.935
SECTION 4.4
1. f(x) = 12(x + 2)−1/2, x >−2;
f (−2) = 0 endpt and abs min; as x→ ∞, f(x) → ∞; so no abs max
2. f(x) = 2x− 3; critical pt. 32; f3
2
=−14 local and abs min
3. f(x) = 2x− 4, x ∈ (0, 3);
critical pt. 2;
f (0) = 1 endpt and abs max, f (2) =−3 local and abs min, f(3) = −2 endpt max 4. f(x) = 4x + 5, x∈ (−2, 0);
critical pt.−54;
f (−2) = −3 endpt max, f
−54
=−338 local and abs min, f (0) =−1 endpt and abs max
5. f(x) = 2x− 1
x2 =2x3− 1
x2 , x= 0; f(x) = 0 at x = 2−1/3 critical pt. 2−1/3; f(x) = 2 + 2
x3, f
2−1/3
= 6 f
2−1/3
= 2−2/3+ 21/3= 2−2/3+ 2 · 2−2/3= 3 · 2−2/3 local min
6. f(x) = 1− 2 x3 critical pt. 21/3; f
21/3
= 3(2)−2/3 local min
7. f(x) = 2x3− 1 x2 , x∈
1 10, 2
; critical pt. 2−1/3;
f1
10
= 101001 endpt and abs max, f 2−1/3
= 3 · 2−2/3 local and abs min, f (2) = 412 endpt max
SECTION 4.4 141
8. f(x) = 1−x23, x∈ (1,√ 2)
critical pt. 21/3; f (1) = 2 endpt and abs max f
21/3
= 3(2)−2/3 local and abs min, f√ 2
=√
2 + 12 endpt max 9. f(x) = 2x− 3, x ∈ (0, 2);
critical pt. 32;
f (0) = 2 endpt and abs max, f3
2
=−14 local and abs min,
f (2) = 0 endpt max
10. f(x) = 2(x− 1)(x − 2)(2x − 3); x ∈ (0, 4)
critical pts. 1, 32, 2, ; f (0) = 4 endpt max, f (1) = 0 local and abs min, f3
2
=161 local max, f (2) = 0 local and abs min, f (4) = 36 endpt and abs max
11. f(x) =(2− x)(2 + x)
(4 + x2)2 , x∈ (−3, 1);
critical pt. −2;
f (−3) = −133 endpt max, f (−2) = −14 local and abs min, f (1) = 15 endpt and abs max
12. f(x) = 2x
(1 + x2)2, x∈ (−1, 2)
critical pt. 0; f (−1) = 12 endpt max, f (0) = 0 local and abs min, f (2) = 45 endpt and abs max
13. f(x) = 2 (x−√ x)
1− 1
2√ x
, x > 0;
critical pts. 14, 1 ;
f (0) = 0 endpt and abs min, f1
4
=161 local max, f (1) = 0 local and abs min;
as x→ ∞, f(x) → ∞; so no abs max 14. f(x) = 2(2− x2)
(4− x2)1/2, x∈ (−2, 2) critical pts. −√
2,√
2; f (−2) = 0 endpt max, f
−√ 2
=−2 local and abs min, f√
2
= 2 local and abs max, f (2) = 0 endpt min 15. f(x) = 3(2− x)
2√
3− x, x < 3 critical pt. 2;
f (2) = 2 local and abs max, f (3) = 0 endpt min;
as x→ −∞, f(x) → −∞; so no abs min
142 SECTION 4.4
16. f(x) = 12
x−1/2+ x−3/2
, x > 0 no critical pts; no extreme values.
17. f(x) =−13(x− 1)−2/3, x= 1;
endpt and abs max
21. f(x) =−3 sin x
SECTION 4.4 143
25.
f(x) =
⎧⎪
⎪⎨
⎪⎪
⎩
−2, 0 < x < 1 1, 1 < x < 4
−1, 4 < x < 7 critical pts. 1, 4;
f (0) = 0 endpt max, f (1) =−2 local and abs min,
f (4) = 1 local and absolute max, f (7) =−2 endpt and abs min
26. f(x) =
⎧⎪
⎪⎨
⎪⎪
⎩
1, −8 < x < −3 2x + 1, −3 < x ≤ 2 5, 2 < x < 5 critical pts. −3, −12;
f (−8) = 1 endpt min, f(−3) = 6 local max f
−12
=−14 local and abs min
27.
f(x) =
⎧⎪
⎪⎨
⎪⎪
⎩
2x, −2 < x < −1 2− 2x, −1 < x < 3 1, 3 < x < 6 critical pts. −1, 1, 3
f (−2) = 5 endpt max, f(−1) = 2 local and abs min, f (1) = 6 local and abs max, f (3) = 2 local and abs min
28. f(x) =
⎧⎪
⎪⎪
⎪⎨
⎪⎪
⎪⎪
⎩
−2x − 2, −2 < x < 0
−1, 0 < x < 2 1, 2 < x < 3 (x− 2)2, 3 < x < 4 critical pts. −1, 0, 2, 3 ;
f (−2) = 2 endpt min, f(−1) = 3 local and abs max, f (0) not an extreme value, f (2) = 0 local and abs min, f (3) = 13 local min, f (4) = 83 endpt max
144 SECTION 4.4
29.
f(x) =
⎧⎪
⎪⎪
⎪⎨
⎪⎪
⎪⎪
⎩
−1, −3 < x < −1 1, −1 < x < 0 2x− 4, 0 < x < 3 2, 3≤ x < 4 critical pts.−1, 0, 2
f (−3) = 2 endpt and abs max, f(−1) = 0 local min, f (0) = 2 local and abs max, f (2) =−2 local and abs min
30. f(x) =
⎧⎪
⎪⎨
⎪⎪
⎩
−2x, 0 < x < 1
−2, 1 < x < 2
−x, 2 < x < 3 Note: 1 is not in the domain of f . critical pts. 2;
f (0) = 0 endpt and abs max, f (2) =−2 local max, f (3) =−92 endpt and abs min
31. 32.
33. Not possible: f (1) = f (3) = 0 implies f(c) = 0 for some c∈ (1, 3) (Rolle’s theorem).
34. f (x) = x− 1
2πsin 2πx
35. Let p(x) = x3+ ax2+ bx + c. Then p(x) = 3x2+ 2ax + b is a quadratic with discriminant Δ = 4a2− 12b = 4(a2− 3b). If a2≤ 3b, then Δ ≤ 0. This implies that p(x) does not change sign on (−∞, ∞).
On the other hand, if a2− 3b > 0, then Δ > 0 and p has two real zeros, c1 and c2, from which it follows that p has extreme values at c1 and c2. Therefore, if p has no extreme values, then we must have a2− 3b ≤ 0.
SECTION 4.4 145
36. f (x) = (1 + x)r− (1 + rx), x ≥ −1.
f(x) = r
(1 + x)r−1− 1
; f(x) = 0 =⇒ x = 0
f(x) = r(r− 1)(1 + x)r−2; f(0) = r(r− 1) > 0 =⇒ f has a local minimum at x = 0 Since 0 is the only critical number of f , the local minimum must be an absolute minimum.
37. By contradiction. If f is continuous at c, then, by the first-derivative test (4.3.4), f (c) is not a local maximum.
38. Since f (c)≥ f(x) for all x in some open interval around c, and likewise f(c) ≤ f(x) for all x in some open interval around c, it follows that f must be constant on some open interval containing c.
39. If f is not differentiable on (a, b), then f has a critical point at each point c in (a, b) where f(c) does not exist. If f is differentiable on (a, b), then by the mean-value theorem there exists c in (a, b) where f(c) = [f (b)− f(a)]/(b − a) = 0. This means c is a critical point of f.
40. We give a proof by contradiction. Suppose for no c in (c1, c2) is f (c) a local minimum. By Theorem 2.6.2, f has a minimum on [c1, c2] and therefore, this minimum must occur at c1 or c2. Suppose that f (c1) is an endpoint minimum. Then for some δ1> 0,
(*) f (x)≥ f(c), x ∈ [c1, c1+ δ1).
Since f (c1) is a local maximum, there exists δ2> 0 such that
(**) f (x)≤ f(c1), x∈ (c1− δ2, c1+ δ2).
Set δ = min [δ1, δ2]. From (*) and (**), it follows that
f (x) = f (c1), x∈ (c1, c1+ δ).
This means that f has a local minimum on (c1, c2). The argument at c2 is similar.
41. Let f (x) =
⎧⎨
⎩
1, if x is a rational number 0, if x is an irrational number
42. f (x) = sin x and f (x) = cos x each have an infinite number of local maxima and local minima occur-ring at distinct points. However, the local maximum values are all the same, 1, and the local minimum values are all the same, −1. The function f(x) = 12x + sin x has an infinite number of local maxima and local minima, and the local maximum values and the local minimum values are all different.
43. Let M be a positive number. Then P (x)− M ≥ anxn−
|an−1|xn−1+· · · + |a1|x + |a0| + M
for x > 0
≥ anxn− xn−1(|an−1| + · · · + |a1| + |a0| + M) for x > 1
≥ xn−1[anx− (|an−1| + · · · + |a1| + |a0| + M)]
It now follows that
P (x)− M ≥ 0 for x≥ K = |an−1| + · · · + |a1| + |a0| + M an
1/n
.