Corollary 6.5.13: Finding Local Extrema

Suppose f is differentiable on (a, b). Then:

If f'(c) = 0 and f'(x) > 0 on (a, x) and f'(x) < 0 on (x, b), then f(c) is a local minimum.
If f'(c) = 0 and f'(x) < 0 on (a, x) and f'(x) > 0 on (x, b), then f(c) is a local maximum.

Context

Proof:

This corollary becomes obvious when we interpret what it means for the function to have a positive or negative derivative, as in these tables:

Loc. Max
interval	(a, c)	(c , b)
sign of f'(x)	+	-
dir. of f(x)	up	down

Loc. Min
interval	(a, c)	(c , b)
sign of f'(x)	-	+
dir. of f(x)	down	up

No Extremum
interval	(a, c)	(c , b)
sign of f'(x)	+	+
dir. of f(x)	up	up

No Extremum
interval	(a, c)	(c , b)
sign of f'(x)	-	-
dir. of f(x)	down	down

Of course, these tables are no proof - which is once again left as an exercise.

Interactive Real Analysis, ver. 1.9.5
(c) 1994-2007, Bert G. Wachsmuth
Page last modified: Mar 28, 2007