If f(x) = 2,g(x) = cos x, then find the point c belongs to0,pi/2 [ which gives the result
of Cauchy's mean value theorem in the interval [0,pi/2] for the functions) fxand
gx.
Answers
Step-by-step explanation:
Does the function satisfy the hypotheses of the Mean Value Theorem on the given interval? f(x) = e^2x, [0, 3] No, is continuous on [0, 3] but not differentiate on (0, 3). Yes, it does not matter if f is continuous or differentiable; every function satisfies the Mean Value Theorem.
Answer:
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
)
.
Step-by-step explanation: