A function $f$ is defined by $f(x)=2+(x-1)^{2 / 3}$ on $[0,2]$. Which of the following is not correct?

$f$ is not derivable in $(0,2)$ $f$ is continuous in $[0,2]$ $f(0)=f(2)$ Rolle's theorem is applicable on $[0,2]$

Rolle's theorem is applicable on $[0,2]$

We have, $f(x)=2+(x-1)^{2 / 3} \Rightarrow f^{\prime}(x)=\frac{2}{3(x-1)^{1 / 3}}$ Clearly, $f(x)$ is not differentiable at $x=1$. So, Rolle's theorem is not applicable on $[0,2]$.