Answer
$f(x)$ is not continuous over the closed interval $[0,1]$,
so the theorem does not apply.
Work Step by Step
Rolle's Theorem requires
- that $f(x)$ is continuous on $[a,b]$
- that $f'(x)$ is defined on $(a,b)$
Observing the right endpoint of $[0,1],\quad $
$f(1)=0$
but
$\displaystyle \lim_{x\rightarrow 1^{-}}f(x)=1$,
so f is not left-continuous at the right endpoint.
This means that f is not continuous over the CLOSED interval $[0,1]$, so the theorem does not apply.