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Byju's Answer
Standard XII
Mathematics
Relation between Continuity and Differentiability
Let fx = x|x|...
Question
Let
f
(
x
)
=
x
|
x
|
,
g
(
x
)
=
sin
(
x
)
and
h
(
x
)
=
(
g
∘
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 continuous but not differentiable at
x
=
0.
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D
h
′
(
x
)
is differentiable at
x
=
0.
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Solution
The correct option is
C
h
′
(
x
)
is continuous but not differentiable at
x
=
0.
f
(
x
)
=
x
|
x
|
f
(
x
)
=
{
x
2
;
x
≥
0
−
x
2
;
x
<
0
h
(
x
)
=
(
g
∘
f
)
(
x
)
=
sin
(
x
|
x
|
)
∀
x
∈
R
h
(
x
)
=
{
sin
(
x
2
)
;
x
≥
0
−
sin
(
x
2
)
;
x
<
0
h
′
(
x
)
=
{
2
x
cos
(
x
2
)
;
x
≥
0
−
2
x
cos
(
x
2
)
;
x
<
0
L.H.L
=
lim
x
→
0
−
−
2
x
cos
(
x
2
)
=
0
R.H.L
=
lim
x
→
0
+
2
x
cos
(
x
2
)
=
0
⇒
L.H.L
=
R.H.L
∴
h
′
(
x
)
is continous at
x
=
0
L.H.D
=
lim
x
→
0
−
h
′
(
x
)
−
h
′
(
0
)
x
−
0
=
lim
x
→
0
−
−
2
x
cos
(
x
2
)
−
0
x
=
lim
x
→
0
−
−
2
cos
(
x
2
)
=
−
2
R.H.D
=
lim
x
→
0
+
h
′
(
x
)
−
h
′
(
0
)
x
−
0
=
lim
x
→
0
+
2
x
cos
(
x
2
)
−
0
x
=
lim
x
→
0
+
2
cos
(
x
2
)
=
2
⇒
L.H.D
≠
R.H.D
∴
h
′
(
x
)
is not differentiable at
x
=
0
Suggest Corrections
0
Similar questions
Q.
Let
f
(
x
)
=
x
|
x
|
,
g
(
x
)
=
sin
x
and
h
(
x
)
=
(
g
o
f
)
(
x
)
. Then
Q.
let f(x)=sinx , g(x)=[x+1] and g(f(x))=h(x) then
h
′
(
π
2
)
Q.
Let
f
(
x
)
=
2
x
−
sin
x
and
g
(
x
)
=
3
√
x
, then
Q.
Let
f
(
x
)
=
[
x
[
x
]
]
,
g
(
x
)
=
[
x
[
1
x
]
]
and
h
(
x
)
=
[
[
x
]
x
]
,
, then
lim
x
→
2
−
f
(
x
)
+
lim
x
→
1
2
+
g
(
x
)
+
lim
x
→
2
+
h
(
x
)
=
Q.
Assertion :Let
f
(
x
)
=
tan
x
and
g
(
x
)
=
x
2
then
f
(
x
)
+
g
(
x
)
is neither even nor odd function. Reason: If
h
(
x
)
=
f
(
x
)
+
g
(
x
)
, then
h
(
x
)
does not satisfy the condition
h
(
−
x
)
=
h
(
x
)
and
h
(
−
x
)
=
−
h
(
x
)
.
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