JEE Main & Advanced JEE Main Paper (Held on 10-4-2019 Morning)

  • question_answer Let \[f:R\to R\]be differentiable at \[c\in R\]and \[f(c)=0\]. If \[g(x)=|f(x)|,\]then at \[x=c,g\]is : [JEE Main 10-4-2019 Morning]

    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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