Corollary: Finding Local Extrema

• Suppose f is differentiable on (a, b). Then:
  1. 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.
  2. 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.

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