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Byju's Answer
Standard XII
Mathematics
Theorems for Differentiability
Verify Langra...
Question
Verify Langrange's mean value theorem for the function:
f
(
x
)
=
x
(
1
−
log
x
)
and find the value of '
c
' in the interval [1, 2].
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Solution
f
(
x
)
=
x
(
1
−
log
x
)
The function
f
(
x
)
is continuous in
[
1
,
2
]
because
x
is continuous every where while
log
x
is continuous when
x
ϵ
[
1
,
∞
]
The function
f
(
x
)
is also differential in
(
1
,
2
)
Therefore there exist
c
′
, such that
c
belongs to
(
1
,
2
)
f
′
(
c
)
=
f
(
b
)
−
f
(
a
)
b
−
a
f
′
(
c
)
=
f
(
2
)
−
f
(
1
)
2
−
1
f
′
(
c
)
=
(
2
−
2
log
2
)
−
(
1
−
log
1
)
2
−
1
f
′
(
c
)
=
1
−
2
log
2
2
−
1
f
′
(
x
)
=
−
log
x
f
′
(
c
)
=
−
log
c
−
log
c
=
1
−
2
log
2
log
c
=
2
log
2
−
1
log
c
=
log
4
−
1
log
4
−
log
c
=
1
log
4
c
=
1
4
c
=
e
1
c
=
4
e
1
c
=
1.472
Thus the value of
c
belongs to
(
1
,
2
)
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0
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