Verify Cauchy's Mean Value Theorem for the functions () = 3() = 3 in the
interval [−
6
, 0], and define Maclaurin's infinite series expansion of the function log(1 + 2).
Answers
Answer:
Cauchy’s Mean Value Theorem generalizes Lagrange’s Mean Value Theorem. This theorem is also called the Extended or Second Mean Value Theorem. It establishes the relationship between the derivatives of two functions and changes in these functions on a finite interval.
French mathematician Augustin-Louis Cauchy (1789-1857)
Fig.1 Augustin-Louis Cauchy (1789-1857)
Let the functions
f
(
x
)
and
g
(
x
)
be continuous on an interval
[
a
,
b
]
,
differentiable on
(
a
,
b
)
,
and
g
′
(
x
)
≠
0
for all
x
∈
(
a
,
b
)
.
Then there is a point
x
=
c
in this interval such that
f
(
b
)
−
f
(
a
)
g
(
b
)
−
g
(
a
)
=
f
′
(
c
)
g
′
(
c
)
.
Proof.
First of all, we note that the denominator in the left side of the Cauchy formula is not zero:
g
(
b
)
−
g
(
a
)
≠
0.
Indeed, if
g
(
b
)
=
g
(
a
)
,
then by Rolle’s theorem, there is a point
d
∈
(
a
,
b
)
,
in which
g
′
(
d
)
=
0.
This, however, contradicts the hypothesis that
g
′
(
x
)
≠
0
for all
x
∈
(
a
,
b
)
.
We introduce the auxiliary function
F
(
x
)
=
f
(
x
)
+
λ
g
(
x
)
and choose
λ
in such a way to satisfy the condition
F
(
a
)
=
F
(
b
)
.
In this case we get
f
(
a
)
+
λ
g
(
a
)
=
f
(
b
)
+
λ
g
(
b
)
,
⇒
f
(
b
)
−
f
(
a
)
=
λ
[
g
(
a
)
−
g
(
b
)
]
,
⇒
λ
=
−
f
(
b
)
−
f
(
a
)
g
(
b
)
−
g
(
a
)
.
and the function
F
(
x
)
takes the form
F
(
x
)
=
f
(
x
)
−
f
(
b
)
−
f
(
a
)
g
(
b
)
−
g
(
a
)
g
(
x
)
.
This function is continuous on the closed interval
[
a
,
b
]
,
differentiable on the open interval
(
a
,
b
)
and takes equal values at the boundaries of the interval at the chosen value of
λ
.
Then by Rolle’s theorem, there exists a point
c
in the interval
(
a
,
b
)
such that
F
′
(
c
)
=
0.
Hence,
f
′
(
c
)
−
f
(
b
)
−
f
(
a
)
g
(
b
)
−
g
(
a
)
g
′
(
c
)
=
0
or
f
(
b
)
−
f
(
a
)
g
(
b
)
−
g
(
a
)
=
f
′
(
c
)
g
′
(
c
)
.
By setting
g
(
x
)
=
x
in the Cauchy formula, we can obtain the Lagrange formula:
f
(
b
)
−
f
(
a
)
b
−
a
=
f
′
(
c
)
.
Cauchy’s mean value theorem has the following geometric meaning. Suppose that a curve
γ
is described by the parametric equations
x
=
f
(
t
)
,
y
=
g
(
t
)
,
where the parameter
t
ranges in the interval
[
a
,
b
]
.
When changing the parameter
t
,
the point of the curve in Figure
2
runs from
A
(
f
(
a
)
,
g
(
a
)
)
to
B
(
f
(
b
)
,
g
(
b
)
)
.
According to the theorem, there is a point
(
f
(
c
)
,
g
(
c
)
)
on the curve
γ
where the tangent is parallel to the chord joining the ends
A
and
B
of the curve.