question_answer

If \[f(x)=\left\{ \begin{matrix}    {{x}^{3}},\,\,{{x}^{2}}<1  \\    x,\,\,\,{{x}^{2}}>1  \\ \end{matrix} \right.\], then \[f(x)\] is differentiable at

A) \[(-\infty ,\infty )-\{1\}\]  

B) \[(-\infty ,\infty )\tilde{\ }\{1,-1\}\]

C) \[(-\infty ,\infty )\tilde{\ }\{1,-1,0\}\]

D) \[(-\infty ,\infty )\tilde{\ }\{-1\}\]

Correct Answer: B

Solution :

[b] \[f(x)\]is clearly continuous for\[x\in R\]. \[f'(x)\]is non-differentiable at \[x=1,\,\,-1.\] \[f'(x)=\left\{ \begin{matrix}    3{{x}^{2}},\,\,\,{{x}^{2}}<1  \\    1,\,\,\,\,\,\,\,\,\,{{x}^{2}}>1  \\ \end{matrix} \right.\] Thus, f(x) is non-differentiable at \[x=1,\,-1\]