5128_ch04ansTE_pp652-661

656 Additional Answers
53.
y
4
3
2
1
y = f(x)
By the First Derivative test, f has relative minima at x 1 and x 4
and relative maxima at x 2.
(b) f (x) 6(x2 )
(x
2 2 2x(6x)
2) 2 6(x
2 2)
(x 2 .
2) 2 Since f changes sign at
x 2, there are points of inflection at x 2 .
(c) Comparing the two relative maxima, we see that f(2) 3 ln 6 4
and f(2) 3ln 6 4. The absolute maximum is 3 ln 6 4.
54.
–1
1
2
3
4
5
6
x
Section 4.4
Exercises 4.4
18. (a) V(x) 2x 3 25x 2 75x
(b) Domain: (0, 5)
61. (a) In Exercise 7, a 4 and b 21,
b
so 7 ,which is the x-value where the point of inflection occurs.
3 a 4
The local extrema are at x 2 and x 3 ,which are symmetric
2
about x 7 4 .
b
(b) In exercise 2, a 2 and b 6, so 1, which is the x-value
3 a
where the point of inflection occurs. The local extrema are at x 0 and
x 2, which are symmetric about x 1.
(c) f (x) 3ax 2 2bx c and f (x) 6ax 2b.
The point of inflection will occur where f (x) 0, which is at
b
x .
3 a
If there are local extrema, they will occur at the zeros of f (x). since
f (x) is quadratic, its graph is a parabola and any zeros will be
symmetric about the vertex which will also be where f (x) 0.
abce 62. (a) f (x) b x
(e bx
,
a) 2 so the sign of f (x) is the
same as the sign of the product abc.
(b) f (x) ab2 ceb
x ( e bx a)
( e bx
.
a)
3 Since a 0, this changes sign
when x ln a
due to the e bx a factor in the numerator, and
b
there is a point of inflection at that location.
63. (a) Since f (x) is quadratic it must have 0, 1, or 2 zeros. If f (x) has 0 or 1
zeros, it will not change sign and the concavity of f(x) will not change,
so there is no point of inflection. If f (x) has 2 zeros, it will change
sign twice, and f(x) will have 2 points of inflection.
(b) f(x) has two points of inflection if and only if 3b 2 8ac.
Quick Quiz (Sections 4.1–4.3)
6x
4. (a) f(x) x 2 2 2(x 1)( x 2)
2
x 2 . Critical numbers are x 1
2
and x 2. The sign graph of f is shown below.
–2 1 2 4
x
(c) Maximum volume 66.02 in 3 when x 1.96 in.
(d) V (x) 6x 2 50x 75, so the critical point is at x 25 57
,
6
which confirms the result in part (c).
19. (a) V(x) 2x(24 2x)(18 2x)
(b) Domain: (0, 9)
[0, 9] by [–400, 1600]
(c) Maximum volume 1309.95 in 3 when x 3.39 in.
(d) V(x) 24x 2 336x 864, so the critical point is at x 7 13,
which confirms the result in part (c).
(e) x 2 in. or x 5 in.
(f) The dimensions of the resulting box are 2x in., (24 2x) in.,
and (18 2x) in. Each of these measurements must be positive, so that
gives the domain of (0, 9).
22. Dimensions: Radius 10
2 3 20
8.16 cm, height 11.55 cm;
3
4000
maximum volume 2418.40 cm 3
33
23. Set r(x) c(x): 4x 1/2 4x. The only positive critical value is x 1, so
profit is maximized at a production level of 1000 units. Note that
(r c)(x) 2(x) 3/2 4 0 for all positive x, so the Second Derivative
Test confirms the maximum.
24. Set r(x) c(x): 2x/(x 2 1) 2 (x 1) 2 . We solve this equation graphically
to find that x 0.294 or x 1.525. The graph of y r(x) c(x)
shows a minimum at x 0.294 and a maximum at x 1.525, so profit is
maximized at a production level of about 1,525 units.
25. Set c(x) c(x)/x:3x 2 20x 30 x 2 10x 30. The only positive
solution is x 5, so average cost is minimized at a production level of
d 2
5000 units. Note that d x2 c( x) 2 0 for all positive x, so the Second
x
Derivative Test confirms the minimum.
26. Set c(x) c(x) /x: xe x e x 4x e x 2x. The only positive solution is
x ln 2, so average cost is minimized at a production level of 1000 ln 2,
d 2
which is about 693 units. Note that d x2
c( x)
x
e x 0 for all positive x,
so the Second Derivative Test confirms the minimum.