2 marks
Q.No : 6
If f(x)=x² and g(x)=x on [a, b], then c of CMVT is
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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
)
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