6.5. Differentiable Functions

Corollary 6.5.13: 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 maximum.
  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 minimum.

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:

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