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Byju's Answer
Standard XII
Mathematics
Derivative of Standard Functions
Let fx=x|x|...
Question
Let
f
(
x
)
=
x
|
x
|
,
g
(
x
)
=
sin
x
and
h
(
x
)
=
(
g
o
f
)
(
x
)
. Then
A
h
(
x
)
is not differentiable at
x
=
0
.
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B
h
(
x
)
is differentiable at
x
=
0
, but
h
′
(
x
)
is not continuous at
x
=
0
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C
h
′
(
x
)
is differentiable at
x
=
0
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D
h
′
(
x
)
is continuous at
x
=
0
but it is not differentiable at
x
=
0
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Solution
The correct option is
C
h
′
(
x
)
is differentiable at
x
=
0
We have,
f
(
x
)
=
−
x
2
,
x
<
0
f
(
x
)
=
x
2
,
x
≥
0
g
(
x
)
=
sin
x
Hence,
h
(
x
)
=
g
(
f
(
x
)
)
=
sin
(
−
x
2
)
,
x
<
0
h
(
x
)
=
g
(
f
(
x
)
)
=
sin
(
x
2
)
,
x
≥
0
As
x
→
0
,
h
(
x
)
→
0
.
h
(
0
)
=
0
.
Hence,
h
(
x
)
is continuous at
x
=
0
.
For
h
(
x
)
, at
x
=
0
,
L.H.D
=
−
2
x
×
cos
(
−
x
2
)
=
0
R.H.D
=
2
x
×
cos
(
x
2
)
=
0
Hence,
h
(
x
)
is differentiable at
x
=
0
.
We have,
h
′
(
x
)
=
−
2
x
cos
(
−
x
2
)
,
x
<
0
h
′
(
x
)
=
2
x
cos
(
x
2
)
,
x
≥
0
Let us consider the derivative of
h
′
(
x
)
at
x
=
0
,
L.H.D
=
−
2
cos
(
−
x
2
)
+
4
x
2
sin
(
−
x
2
)
=
−
2
R.H.D
=
2
cos
(
x
2
)
−
4
x
2
sin
(
x
2
)
=
2
Hence,
h
′
(
x
)
is not differentiable at
x
=
0
.
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0
Similar questions
Q.
Let
f
(
x
)
=
x
|
x
|
,
g
(
x
)
=
sin
(
x
)
and
h
(
x
)
=
(
g
∘
f
)
(
x
)
. Then
Q.
The function
f
(
x
)
=
x
tan
−
1
1
x
for
x
≠
0
,
f
(
0
)
=
0
is :
Q.
Let
f
(
x
)
=
{
x
n
sin
1
x
,
x
≠
0
0
,
x
=
0
, then f(x) is continuous but not differentiable at x=0 if
Q.
If
f
(
x
)
=
{
x
sin
1
x
else where
0
x
=
0
,
then
f
(
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)
is
Q.
If
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x
)
=
⎧
⎨
⎩
x
e
−
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1
|
x
|
+
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x
)
,
i
f
x
≠
0
0
,
i
f
x
=
0
then f(x) is
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