## 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 function cuts the -axis there must be a point where The following picture illustrates the theorem. Rolle: somewhere between a and b , the graph must be flat

Like most important theorems, Rolle's theorem has to be stated rather carefully in order to make sure that it is true:

Suppose that is a differentiable function whose derivative is a continuous function. Suppose that and (with let's say). Then there must be at least one point between and ( ) at which (The precise behaviour of outside the interval is 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 points and in the interval with the property that for every x in the interval [ a , b ]. In particular, this property implies at the end points that If then since cannot be an end-point. But then since otherwise, an interior minimum would not occur at Thus the conclusion holds with Now if then either or . In the first case implies and hence else would not yield an interior maximum of In this first case, the conclusion holds with Now in the remaining second case In this case, the function is constant on the interval Thus for every point in the interval. The conclusion is obvious.

Note that the function f ( x ) in the above discussion could have several minima and maxima interior to the interval. At these points the slope or derivative f ¢ ( c ) = 0. 