Corollary 6.5.13: Finding Local Extrema
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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 maximum.
- 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.
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
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