Can Rolle's Theorem be true for the critical point where derivative doesnt exist? there is the problem that I met



At 0 the derivative of f(x) doesn't exist so 0 is the critical number but the conclusion of Rolle's theorem is the f'(c) (here c=0) must be 0. Are there any explaination in this case
A critical point is a point where the derivative exists and equals zero. So $c=0$ is not a critical point of $f$. The point here is that $f$ fails to satisfy all of the assumptions of Rolle's theorem, and indeed the conclusion fails, too.
If f is differentiable, it also be continuous and have critical point. According to Fermat theorem, if f is differentiable at this point it's derivative must be 0. So your example does not meet the prerequisite.