On-line Math 21

4.1  The Mean Value Theorem

Theorem 6. [Mean Value Theorem] If f is continuous on a closed interval [a,b] , and differentiable on the open interval (a,b) , then there is some point c Î (a,b) so that

f¢(c) = f(b)-f(a) b-a .

Proof. The (venerable) trick is to construct from f another function, one which satisfies Rolle's theorem, and see what it tells us. Set

g(x): = f(x)-f(a)- f(b)-f(a) b-a (x-a).

Then, because f is, g is continuous on the closed interval [a,b] , and differentiable on the open interval (a,b) . Also, g(a) = g(b) = 0 , so we satisfy Rolle's theorem's hypothesis. That means that there is some c Î (a,b) so that g¢(c) = 0 . But,

g¢(x) = f¢(x)- f(b)-f(a) b-a ,

so

0 = g¢(c) = f¢(c)- f(b)-f(a) b-a Þ f¢(c) = f(b)-f(a) b-a ,

as desired.

Copyright (c) 2000 by David L. Johnson.