On-line Math 21

4.2  Corollaries of the Mean Value Theorem

Corollary 3 If f is continuous on [a,b] , and has a positive derivative ( f¢(x) > 0 ), on (a,b) , then f is increasing on the interval (a,b) . If f¢(x) < 0 for all x Î [a,b] , then f is decreasing on (a,b) .

Proof. Since f¢(x) > 0 on this interval, then for any x < y Î (a,b) , there is a c Î (x,y) so that

0 < f¢(c) = f(y)-f(x) y-x ,

so f(y)-f(x) > 0 , or f(y) > f(x) , that is, the function is increasing. As usual, turn the inequalities around for the opposite direction.

Copyright (c) 2000 by David L. Johnson.