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.

\includegraphics[width=8cm]{xfig/rolle.eps}

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 thatis a differentiable function whose derivative is a continuous function. Suppose thatand(withlet's say). Then there must be at least one point betweenand( ) at which

(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 for every x in the interval [ a , b ]. In particular, this property implies at the end points thatIfthensincecannot be an end-point. But thensince otherwise, an interior minimum would not occur at Thus the conclusion holds withNow ifthen eitheror. In the first caseimplies and henceelsewould not yield an interior maximum ofIn this first case, the conclusion holds withNow in the remaining second caseIn this case, the functionis constant on the intervalThusfor every pointin 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.