On-line Math 21

4.1  The Mean Value Theorem

Theorem 2 If f(x) has a local extremum at c , and if f¢(c) exits, then f¢(c) = 0 .

Proof. Assume that f has a maximum at c , and assume that f¢(c) exists. Could f¢(c) be positive? No, because if so, since

f¢(c) = lim h® 0  f(c+h)-f(c) h ,

if f¢(c) > 0 , then for close enough h , with h > 0 ,

f(c+h)-f(c) h > 0,

which says that, in particular, f(c+h) > f(c) . So, since f(c) is the maximum, this couldn't be. Similarly, if f¢(c) < 0 , then (take h < 0 this time) again f(c) could not have been the largest.
Since both of these cases are impossible, the only remaining case, f¢(c) = 0 , must be true.

Copyright (c) 2000 by David L. Johnson.