## Rolle's Theorem

Rolle's theorem is an important basic result about differentiable functions. Like many basic results in the calculus it seems very obvious. It just says that between any two points where the graph of the differentiable functioncuts the * -axis there must be a point whereThe following picture illustrates the theorem.*

Rolle: somewhere between
Like most important theorems, Rolle's theorem has to be stated rather carefully in order to make sure that it is true: Suppose thatis a differentiable function whose derivative is a continuous function. Suppose thatand(withlet's say). Then there must be at least one point (The precise behaviour ofoutside the intervalis not really relevant and the theorem can be stated in a more general form.) Proof:In general, the extreme value theorem for continuous functions implies there exists at least two pointsandin the intervalwith the property that Note that the function |