Rolle's Theorem is a special case of the Mean
Value Theorem which says that there has to be a point
between $a$ and $b$ where the instantaneous
rate of change is equal to
the average rate of change between $a$ and
$b$. More precisely:
Mean Value
Theorem: If $f$ is a function that is continuous on
the closed interval $[a,b]$ and differentiable on the open
interval $(a,b)$, then there is a point $c$ in $(a,b)$
such that $\displaystyle f'(c)=\frac{f(b)-f(a)}{b-a}$.