Chapter 5
Rolle’s Theorem, the Mean Value
Theorem, and L’Hˆopital’s Rule
5.1
Rolle’s Theorem
In the following problems
(a) Verify that the three conditions of Rolle’s theorem have been met.
(b) Find all values
z
that satisfy the conclusion of the theorem.
1. (a)
f
(
x
)=
x
2

7
x
+10
,
on
[2
,
5]
(b)
f
(
x
x
2

7
x
,
on
[0
,
7]
(c)
f
(
x
x
2

7
x
,
on
[

1
,
8]
2. (a)
f
(
x
x
2

4
x,
on
[0
,
4]
(b)
f
(
x
x
2

4
x,
on
[

1
,
5]
(c)
f
(
x
x
2

4
x,
on
[

4
,
8]
3. (a)
f
(
x
x
3

5
x
2

17
x
+21
,
on

3
≤
x
≤
7
(b)
f
(
x
x
3

16
x,
on

4
≤
x
≤
4
(c)
f
(
x
x
3
+2
x
2

x

2
,
on

2
≤
x
≤
1
4. (a)
f
(
x
x
2

6
x

7
,
on
[

1
,
7]
(b)
f
(
x
x
3
+
x
2

6
x,
on
[0
,
2]
(c)
f
(
x
x
2

x

2
,
on
[

1
,
2]
(d)
g
(
x
x
3
+5
x
2
+6
,
on
[

3
,
0]
5. (a)
f
(
x
)=s
in2
x,
on
[0
,π
]
(b)
f
(
x
) = cos
(
x
2
)
,
on
[
π,
3
π
]
(c)
f
(
x
in
x
+ cos
x
on
£

π
4
,
3
π
4
/
(
Hint:
3
π
4
=
π
4
+
π
2
)
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