As Rolle theorem goes,if
- $f(x)$ is continuous and well-defined in $[a,b]$,
- derivable in $(a,b)$, and $f'(x)$ is bounded,
- $f(a)=f(b)$,
then there exists $c$ ($a<c<b$), which satisfies $f'(c)=0$.
I wonder how to find how many $c$'s there are in a given interval like $[a,b]$? e.g. while seeking the number of roots of Legendre multinomial, although $x=1$ and $x=-1$ are two null points, we can't say there only exits one $c$ ($f'(c)=0$).
Rolle's theorem is what mathematicians call an "existence proof", which proves that something exists without actually telling you how to calculate it. The opposite of that is (according to Wikipedia) a "constructive proof" or an "effective proof".
Solving for $f'(x)=0$ is described as finding the critical points of the function $f$. Critical points may be either local minima, local maxima, or points like the one at $x=0$ for $f(x)=x^3$ where the graph is just parallel with the $x$ axis. We have to also compute $f''(x)$ to work out which one.
Rolle's theorem just tells you that a critical point (in fact a maximum or a minimum, since $f(a)=f(b)$) must exist, without telling you how to calculate it. There is no general way to tell how many exist. For example for the case that $f(x)=C$, where $C$ is a constant, it's clear that $f'(x)=0$ for all values of $x\in [a,b]$.