Corollary 1 If f¢(x) = 0 for all x Î [a,b] , then f is a constant.
Proof. If f(x) were not constant, there would be a point x Î (a,b) where f(x) ¹ f(a) . Say, for instance, that f(x) > f(a) . Then, by MVT, there is a point c Î (a,x) , where f¢(c) = (f(x)-f(a)/(x-a) ¹ 0 , which contradicts the fact that f¢(x) = 0 for all x . On the other hand, if f(x) < f(a) , the same argument (with the signs reversed), shows that again a contradiction is reached. The only other case is that f(x) is never unequal to f(a) .
Copyright (c) 2000 by David L. Johnson.