162
Section 4.1
11. (a)
d
dx
xxx
()
32
232
−=−
′
=−
=
f
13
1 21
2
(b)
d
dx
x
+=
21
′
=
f
31
(c)
Left-hand derivative:
lim
(
)
[(
)
(
)]
hh
fh
f
h
→
−
→
−
+−
=
+−+−
00
3
22
2
2
2
4
61
0
0
10
0
0
2
h
h
h
h
h
=
++
=+
+
=
→
−
→
−
lim (
)
Right-hand derivative:
(
)
)
]
f
h
h
h
→
+
→
+
=
++−
=
2
2
4
h
h
h
h
→
+
→
+
=
=
0
0
1
1
Since the left-and right-hand derivatives are not equal,
′
f
2 is underfined.
12. (a)
The domain is
x
≠
2. (See the solution for 11.(c)).
(b)
′
=
−<
>
⎧
⎨
⎩
fx
xx
x
,
,
2
12
2
Section 4.1
Exercises
1.
Minima at (–2, 0) and (2, 0), maximum at (0, 2)
2.
Local minimum at (–1, 0), local maximum at (1, 0)
3.
Maximum at (0, 5) Note that there is no minimum since the
endpoint (2, 0) is excluded from the graph.
4.
Local maximum at (–3, 0), local minimum at (2, 0),
maximum at (1, 2), minimum at (0, –1)
5.
Maximum at
x
=
b
, minimum at
xc
=
2
;
The Extreme Value Theorem applies because
f
is continuous
on [
a
,
b
], so both the maximum and minimum exist.
6.
Maximum at
x
=
c
, minimum at
x
=
b
;
The Extreme Value Theorem applies because
f
is continuous
on [
a
,
b
], so both the maximum and minimum exist.