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Question

If $$f(x)$$ is differentiable everywhere, then


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

The correct option is D $${ \left| f\left( x \right) \right|  }^{ 2 }$$ is differentiable everywhere
If $$f(x)$$ is differentiable, $${ \left[ f\left( x \right) \right]  }^{ 2 }={ \left| f\left( x \right) \right|  }^{ 2 }$$ is alos differentiable.

Taking $$f(x)=x$$, we have $$ f ( x ) =\left| f\left( x \right) \right| =\left| x \right| ,$$ which is not differentiable at $$x=0$$

Thus, $$f$$ is differentiable in $$R\nRightarrow |f|$$ is differentiable in $$R$$.

Also $$\displaystyle \left( f\left| f \right|  \right) \left( x \right) =f\left( x \right) \left| f\left( x \right)  \right| =\begin{cases} \begin{matrix} -{ \left[ f\left( x \right) \right]  }^{ 2 }, & f\left( x \right)<0 \end{matrix} \\ \begin{matrix} { \left[ f\left( x \right) \right]  }^{ 2 }, & f\left( x \right)\ge 0 \end{matrix} \end{cases}$$
which is differentiable in $$R$$ if $$f$$ is differentiable in $$R$$.

Mathematics

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