Answer
(f is differentiable at x=1), $\displaystyle \frac{2}{3}$
(f is not differentiable at x=0)
Work Step by Step
$f^{\prime}(x)=[x-x^{1/3}]^{\prime}=1-\displaystyle \frac{1}{3}x^{-2/3}=1-\frac{1}{3\sqrt[3]{x^{2}}}\\\\$
$a.\displaystyle \ \ \ f^{\prime}(1)=1-\frac{1}{3\sqrt[3]{1^{2}}}=\frac{2}{3}\\\\$
(f is differentiated at x=1)$\\\\$
$b.\ \ \ f^{\prime}(0)$ is not defined (zero in the denominator)$\\\\$
(f is not differentiated at x=0)$\\\\$
