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Byju's Answer
Standard XII
Mathematics
Graphical Interpretation of Differentiability
Let f: R → R ...
Question
Let
f
:
R
→
R
be twice continuously differentiable. Let
f
(
0
)
=
f
(
1
)
=
f
′
(
0
)
=
0
. Then
A
f
′′
(
x
)
≠
0
for all
x
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B
f
′′
(
c
)
=
0
for some
c
∈
R
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C
f
′′
(
x
)
≠
0
if
x
≠
0
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D
f
′
(
x
)
>
0
for all
x
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Solution
The correct option is
B
f
′′
(
c
)
=
0
for some
c
∈
R
Applying Rolle's theorem to
f
(
x
)
on the interval
[
0
,
1
]
, we get
f
′
(
c
1
)
=
0
for some
c
1
∈
(
0
,
1
)
Again, applying Rolle's theorem to
f
′
(
x
)
on the interval
[
0
,
c
1
]
, we get
f
′′
(
c
)
=
0
for some
c
∈
(
0
,
c
1
)
Suggest Corrections
0
Similar questions
Q.
Let f :
R
→
R
be a twice continuously differentiable function such that
f
(
0
)
=
f
(
1
)
=
f
′
(
0
)
=
0
. Then
Q.
Let
f
:
R
→
R
be twice continuously differentiable (or
f
′′
exists and is continuous) such that
f
(
0
)
=
f
(
1
)
=
f
′
(
0
)
=
0
. Then
Q.
Let
f
be a twice differentiable function defined on
R
such that
f
(
0
)
=
1
,
f
′
(
0
)
=
2
and
f
′
(
x
)
≠
0
for all
x
∈
R
.
If
∣
∣
∣
f
(
x
)
f
′
(
x
)
f
′
(
x
)
f
′′
(
x
)
∣
∣
∣
=
0
,
for all
x
∈
R
,
then the value of
f
(
1
)
lies in the interval
Q.
Let
f
:
R
→
R
be a twice continuously differentiable function such that
f
(
0
)
=
f
(
1
)
=
f
′
(
0
)
=
0
. Then
Q.
Let
f
be a twice differentiable function defined on
R
such that
f
(
0
)
=
1
,
f
′
(
0
)
=
2
and
f
′
(
x
)
≠
0
for all
x
∈
R
.
If
∣
∣
∣
f
(
x
)
f
′
(
x
)
f
′
(
x
)
f
′′
(
x
)
∣
∣
∣
=
0
,
for all
x
∈
R
,
then the value of
f
(
1
)
lies in the interval
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