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Question

If f(x)=|log2sinx| and g(x)=f(f(x)), where xR, then

A
g(x) is not differentiable at x=0
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B
g(x) is differentiable at x=0 and g(0)=cos(log2)
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C
g(x) is differentiable at x=0 and g(0)=cos(log2)
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D
g(x) is differentiable at x=0 and g(0)=sin(log2)
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Solution

The correct option is B g(x) is differentiable at x=0 and g(0)=cos(log2)
Given : f(x)=|log2sinx| and g(x)=f(f(x))
In the neighorhood of x=0, sinx<log2, so
f(x)=log2sinxg(x)=log2sinf(x)g(x)=log2sin(log2sinx)
So, g(x) is differentiable at x=0
Now,
g(x)=f(f(x))×f(x)g(0)=f(f(0))×f(0)g(0)=cos(log2)×(cos0)g(0)=cos(log2)

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