A) differentiable if\[f'(c)=0\]
B) not differentiable
C) differentiable if \[f'(c)\ne 0\]
D) not differentiable if \[f'(c)=0\]
Correct Answer: A
Solution :
\[g'(c)=\underset{h\to 0}{\mathop{\lim }}\,\frac{|f(c+h)|-|f(c)|}{h}\] \[=\underset{h\to 0}{\mathop{\lim }}\,\frac{|f(c+h)|}{h}=\underset{h\to 0}{\mathop{\lim }}\,\frac{|f(c+h)-f(c)|}{h}\] \[=\underset{h\to 0}{\mathop{\lim }}\,\left| \frac{f(c+h)-f(c)}{h} \right|\frac{|h|}{h}\] \[=\underset{h\to 0}{\mathop{\lim }}\,|f'(c)|\frac{|h|}{h}=0,\]if\[f'(c)=0\] i.e.,\[g\left( x \right)\]is differentiable at \[x=c,\] if \[f'\left( c \right)=0\]
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