# Let $f:R\rightarrow R$ be differentiable at $c\epsilon R$ and f(c) = 0.If $g(x)=\left | f(x) \right |$, then at x=c , g is : Option 1) not differentiable if $f{}'(c)=0$ Option 2) differentiable if $f{}'(c)\neq0$ Option 3) differentiable if $f{}'(c)=0$ Option 4) not differentiable

V Vakul

Given g(x) = | f(x) | and also given f(c)=0

$g'(c^{+})=\lim_{x\rightarrow c^{+}}\frac{|f(x)|-f(c)}{x-c}$

$=\lim_{x\rightarrow c^{+}}\frac{\pm f(x)}{x-c}$

$=\pm f'(c)$

$g'(c^{-})=\lim_{x\rightarrow c^{-}}\frac{|f(x)|-f(c)}{x-c}$

$=\pm f'(c)$

For  g(x)  to be differentiable at C

$f'(c)=0$

correct option is (3)

Option 1)

not differentiable if $f{}'(c)=0$

Option 2)

differentiable if $f{}'(c)\neq0$

Option 3)

differentiable if $f{}'(c)=0$

Option 4)

not differentiable

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