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The Mean Value Theorem and some of its
consequences.
The mean value theorem is used to justify several basic results in Calculus,
and it also plays a critical role in the Fundamental Theorem of Calculus that
connects differentiation and integration.
We begin with a simplified
version of the mean value theorem.
Rolle's
Theorem: Let
f
be a function that is continuous on a closed interval
[
a
,
b
]
, differentiable on
(
a
,
b
)
, and satisfies
f
(
a
)
=
f
(
b
)
. Then there is a number
c
in
(
a
,
b
)
such that
f
′
(
c
)
=
0
.
The proof of this theorem makes use of the Extreme Value Theorem (which states
that a continuous function defined on a closed interval has both a maximum and
a minimum value). The proof of the Extreme Value Theorem makes use of
properties of the real numbers (the completeness of
ℝ
)
that are outside the scope of a basic calculus course.
The Mean Value Theorem: Let
f
be a function that is continuous on a closed interval
[
a
,
b
]
and differentiable on
(
a
,
b
)
. Then there is a number
c
in
(
a
,
b
)
such that
f
′
(
c
)
=
f
(
b
)
−
f
(
a
)
b
−
a
.
This theorem can be thought of as Rolle's Theorem "on a slope". It is proved
by defining a function
h
(
x
)
that is the difference between
f
(
x
)
and the secant line connecting
(
a
,
f
(
a
)
)
with
(
b
,
f
(
b
)
)
.
This function
h
(
x
)
then satisfies the conditions of Rolle's Theorem, and conclusion that
h
′
(
c
)
=
0
for some
c
in
(
a
,
b
)
leads to the desired conclusion about
f
.
Here are some basic consequences of the Mean Value Theorem.
Corollary: If
f
′
(
x
)
=
0
for all
x
in an interval
(
a
,
b
)
,
then
f
is constant on
(
a
,
b
)
.
Corollary: If
f
′
(
x
)
=
g
′
(
x
)
for all
x
in an interval
(
a
,
b
)
,
then there exists a constant
C
such that
f
(
x
)
=
g
(
x
)
+
C
for all
x
in
(
a
,
b
)
.
Recall that a function
f
is increasing on an interval if for all
x
1
<
x
2
in the interval, it follows that
f
(
x
1
)
<
f
(
x
2
)
.
Likewise
f
is decreasing on an interval if for all
x
1
<
x
2
in the interval, it follows that
f
(
x
1
)
>
f
(
x
2
)
.
Corollary (Increasing/Decreasing Test): If
f
′
(
x
)
>
0
on an interval
(
a
,
b
)
then
f
is increasing on
(
a
,
b
)
.
If
f
′
(
x
)
<
0
on an interval
(
a
,
b
)
then
f
is decreasing on
(
a
,
b
)
.