ROLL'S THEOREM

Rolle's Theorem

Let f be continuous on a closed interval [a, b] and differentiable on the open interval (a, b). If f(a) = f(b), then there is at least one point c in (a, b) where f '(c) = 0.

(The tangent to a graph of f where the derivative vanishes is parallel to x-axis, and so is the line joining the two "end" points (a, f(a)) and (b, f(b)) on the graph. The line that joins to points on a curve -- a function graph in our context -- is often referred to as a secant. Thus Rolle's theorem claims the existence of a point at which the tangent to the graph is parallel to the secant, provided the latter is horizontal.)