A) Continuous for all x but not differentiable for any x
B) Continuous and differentiable for all x
C) Continuous for all x and differentiable for all \[x\ne 0\]
D) Continuous and differentiable for all \[x\ne 0\]
Correct Answer: C
Solution :
[c] \[f(x)=\sqrt[3]{\frac{{{x}^{4}}}{\left| x \right|}},x\ne 0,f(0)=0\] \[\therefore f(x)=\sqrt[3]{\frac{{{x}^{4}}}{-x}}=\sqrt[3]{-{{x}^{3}}}=-x\,\,if\,\,x<0\] \[\And \,f(x)=\sqrt[3]{\frac{{{x}^{4}}}{x}}=\sqrt[3]{{{x}^{3}}}=x\,\,if\,\,x>0\] \[f(x)=\left\{ \begin{matrix} -x, & if & x<0 \\ 0, & if & x=0 \\ x, & if & x>0 \\ \end{matrix} \right.\] Clearly f(x) is continuous for all x but not differentiable at x = 0
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