Rolle's Theorem:
if you have a function, f(x), and you have two points, a and b
(we shall assume that a < b) such that  f(a) = f(b), and both
f(x) and f'(x) exist and are continuous on the entire interval,
a <= x <= b, then there must be a point, c lying between a and b
where  f'(c) = 0.