Corollary 6.5.13: Finding Local Extrema
Why these ads ...
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.