33. (a)
Absolute maximum at (1, 2);
absolute minimum at (3,
2
2)
(b)
None
(c)
One possible answer:
34. (a)
Absolute maximum at (0, 2);
absolute minimum at (2,
2
1) and (
2
2,
2
1)
(b)
At (1, 0) and (
2
1, 0)
(c)
One possible answer:
(d)
Since
f
is even, we know
f
(3)
5
f
(
2
3). By the
continuity of
f
, since
f
(
x
)
,
0 when 2
,
x
,
3, we
know that
f
(3)
#
0, and since
f
(2)
52
1 and
f
9
(
x
)
.
0
when 2
,
x
,
3, we know that
f
(3)
.2
1. In
summary, we know that
f
(3)
5
f
(
2
3),
2
1
,
f
(3)
#
0,
and
2
1
,
f
(
2
3)
#
0.
35.
36.
37. (a)
v
(
t
)
5
s
9
(
t
)
5
2
t
2
4
(b)
a
(
t
)
5
v
9
(
t
)
5
2
(c)
It begins at position 3 moving in a negative direction. It
moves to position
2
1 when
t
5
2, and then changes
direction, moving in a positive direction thereafter.
38. (a)
v
(
t
)
5
s
9
(
t
)
2
2
2
t
(b)
a
(
t
)
5
v
9
(
t
)
2
(c)
It begins at position 6 and moves in the negative
direction thereafter.
39. (a)
v
(
t
)
5
s
9
(
t
)
5
3
t
2
2
3
(b)
a
(
t
)
5
v
9
(
t
)
5
6
t
(c)
It begins at position 3 moving in a negative direction. It
moves to position 1 when
t
5
1, and then changes
direction, moving in a positive direction thereafter.
40. (a)
v
(
t
)
5
s
9
(
t
)
5
6
t
2
6
t
2
(b)
a
(
t
)
5
v
9
(
t
)
5
6
2
12
t
(c)
It begins at position 0. It starts moving in the positive
direction until it reaches position 1 when
t
5
1, and
then it changes direction. It moves in the negative
direction thereafter.
41. (a)
The velocity is zero when the tangent line is horizontal,
at approximately
t
5
2.2,
t
5
6, and
t
5
9.8.
(b)
The acceleration is zero at the inflection points,
approximately
t
5
4,
t
5
8, and
t
5
11.
42. (a)
The velocity is zero when the tangent line is horizontal,
at approximately
t
0.2,
t
5
4, and
t
5
12.
(b)
The acceleration is zero at the inflection points,
approximately
t
5
1.5,
t
5
5.2,
t
5
8,
t
5
11,
and
t
5
13.
43.
No.
f
must have a horizontal tangent at that point, but
f
could be increasing (or decreasing), and there would be no
local extremum. For example, if
f
(
x
)
5
x
3
,
f
9
(0)
5
0 but there is no local extremum at
x
5
0.