24. (a)
f
9
(
x
) is a quadratic polynomial, and as such it can have
0, 1, or 2 zeros. If it has 0 or 1 zeros, then its sign
never changes, so
f
(
x
) has no local extrema.
If
f
9
(
x
) has 2 zeros, then its sign changes twice, and
f
(
x
) has 2 local extrema at those points.
(b)
Possible answers:
No local extrema:
y
5
x
3
;
2 local extrema:
y
5
x
3
2
3
x
25.
Let
x
be the length in inches of each edge of the square
end, and let
y
be the length of the box. Then we require
4
x
1
y
#
108. Since our goal is to maximize volume, we
assume 4
x
1
y
5
108 and so
y
5
108
2
4
x
. The volume is
V
(
x
)
5
x
2
(108
2
4
x
)
5
108
x
2
2
4
x
3
, where 0
,
x
,
27.
Then
V
95
216
x
2
12
x
2
52
12
x
(
x
2
18), so the critical
point occurs at
x
5
18 in. Since
V
9
(
x
)
.
0 for 0
,
x
,
18
and
V
9
(
x
)
,
0 for 18
,
x
,
27, the critical point
corresponds to the maximum volume. The dimensions of
the box with the largest possible volume are
18 in. by 18 in. by 36 in.
26.
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 Spring '08
 GERMAN
 Critical Point

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