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Question

If f(x) is differentiable everywhere then

A
|f(x)| is differentiable everywhere
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B
|f|2 is differatiable everywhere
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C
f|f| is not differentiable at some point
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D
None of these
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Solution

The correct option is A |f|2 is differatiable everywhere
A. Even if f(x) is differentiable everywhere, |f(x)| need not be differentiable everywhere. An example being where the function changes its sign from negative to positive. f(x)=x at x=0

B.
|f|= {fiff>0fiff<0
So,
|f|2= {f2iff>0f2iff<0
Differentiating |f|2,
{2ffiff>02ffiff<0
At f=0,LHD=RHD=|f(0)|2=0, so function is differentiable.

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