Let f : R → R be differentiable at

Question:

Let $f: \mathrm{R} \rightarrow \mathrm{R}$ be differentiable at $\mathrm{c} \in \mathrm{R}$ and $f(c)=0$. If $\mathrm{g}(\mathrm{x})=|f(\mathrm{x})|$, then at $\mathrm{x}=\mathrm{c}, \mathrm{g}$ is :

  1. differentiable if $f^{\prime}(\mathrm{c})=0$

  2. not differentiable

  3. differentiable if $f^{\prime}$ (c) $\neq 0$

  4. not differentiable if $f^{\prime}(\mathrm{c})=0$

Correct Option: 1

Solution:

$g^{\prime}(c)=\lim _{h \rightarrow 0} \frac{|f(c+h)|-|f(c)|}{h}$

$=\lim _{h \rightarrow 0} \frac{|f(c+h)|}{h}=\lim _{h \rightarrow 0} \frac{|f(c+h)-f(c)|}{h}$

$=\lim _{h \rightarrow 0}\left|\frac{f(c+h)-f(c)}{h}\right| \frac{|h|}{h}$

$=\lim _{h \rightarrow 0}\left|f^{\prime}(c)\right| \frac{|h|}{h}=0$, if $f^{\prime}(c)=0$

i.e., $\mathrm{g}(\mathrm{x})$ is differentiable at $\mathrm{x}=\mathrm{c}$, if $\mathrm{f}^{\prime}(\mathrm{c})=0$

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