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 f′(c)=f(b)−f(a)b−a.