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Question

Let f(x)=x|x|,g(x)=sinx and h(x)=(gof)(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)=x2,x<0
f(x)=x2,x0
g(x)=sinx
Hence,
h(x)=g(f(x))=sin(x2),x<0
h(x)=g(f(x))=sin(x2),x0
As x0,h(x)0.
h(0)=0.
Hence, h(x) is continuous at x=0.
For h(x), at x=0,
L.H.D =2x×cos(x2)=0
R.H.D =2x×cos(x2)=0
Hence, h(x) is differentiable at x=0.
We have,
h(x)=2xcos(x2),x<0
h(x)=2xcos(x2),x0
Let us consider the derivative of h(x) at x=0,
L.H.D =2cos(x2)+4x2sin(x2)=2
R.H.D =2cos(x2)4x2sin(x2)=2
Hence, h(x) is not differentiable at x=0.

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