Theorem 1 If f is continuous on a closed interval [a,b] , and differentiable on the open interval (a,b) , and if f(a) = f(b) , then there is some point c Î (a,b) so that f¢(c) = 0 .
If, for some x Î (a,b) , f(x) > f(a) , then there is a maximum point
by the theorem we proved a while back about the existence of a maximum of a
continuous function. That maximum value has to be greater than f(a) ,
so it isn't going to happen at x = b , since f(a) = f(b) . So, it's
at some c Î (a,b) . Our assumption was that the function was differentiable
on (a,b) , so f¢(c) exists. By the previous result, then, f¢(c) = 0 .
The same argument works if, for some x , f(x) < f(a) (all the inequalities are reversed). The only other possibility is that f(x) is never bigger or smaller than f(a) , which says that f(x) is constant, so has 0 derivative at all x Î (a,b) .
Copyright (c) 2000 by David L. Johnson.