Question
If $$f\left( x \right) = \root 3 \of {\frac{{{x^4}}}{{\left| x \right|}}} ,\,x \ne 0$$ and $$f\left( 0 \right) = 0$$ is :
A.
continuous for all $$x$$ but not differentiable for any $$x$$
B.
continuous and differentiable for any $$x$$
C.
continuous for all $$x$$ and differentiable for all $$x \ne 0$$
D.
continuous and differentiable for all $$x \ne 0$$
Answer :
continuous for all $$x$$ and differentiable for all $$x \ne 0$$
Solution :
$$\eqalign{
& f\left( x \right) = \root 3 \of {\frac{{{x^4}}}{{\left| x \right|}}} ,\,x \ne 0,\,f\left( 0 \right) = 0 \cr
& \therefore \,f\left( x \right) = \root 3 \of {\frac{{{x^4}}}{{ - x}}} = \root 3 \of {{ - x^3}} = - x{\text{ if }}x < 0 \cr
& \& \,f\left( x \right) = \root 3 \of {\frac{{{x^4}}}{x}} = \root 3 \of {{x^3}} = x{\text{ if }}x > 0 \cr} $$
\[f\left( x \right) = \left\{ \begin{array}{l}
- x,{\rm{ \,if\, }}x < 0\\
\,\,\,\,\,0,{\rm{ \,if\, }}x = 0\\
\,\,\,\,\,x,{\rm{ \,if\, }}x > 0\,\,
\end{array} \right.\]
Clearly $$f\left( x \right)$$ is continuous for all $$x$$ but not differentiable at $$x = 0$$