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Extreme values
Absolute extreme values
Function f has an absolute maximum value f(x_0) at the point x_0 in its domain if f(x) \leq f(x_0) holds ofr every x in the domain of f.
Similarly, f has an absolute minimum value f(x_1) at the point x_1 in its domain if f(x) \geq f(x_1) holds for every x in the domain of f.
Local extreme values
Function f has an local maximum value f(x_0) at the point x_0 in its domain provided there exists a number h > 0 such that f(x) \leq f(x_0) whenever x is in the domain of f and |x - x_0| < h.
Similarly, f has an local minimum value f(x_1) at the point x_1 in its domain provided there exists a number h > 0 such that f(x) \geq f(x_1) whenever x is in the domain of f and |x - x_1| < h.
Critical points
A critical point is a point x \in \mathrm{Dom}(f) where f'(x) =0.
Singular points
A singular point is a point x \in \mathrm{Dom}(f) where f'(x) is not defined.
Endpoints
An endpoint x \in \mathrm{Dom}(f) that does not belong to any open interval contained in \mathrm{Dom}(f)
Locating extreme values
If the function f is defined on an interval I and has a local maxima or minima in I then the point must be either a critical point of f, a singular point of f or an endpoint of I.
Proof:
Suppose that f has a local maximum value at x_0 and that x_0 is neither an endpoint of the domain of f nor a singular point of f. Then for some h > 0, f(x) is defined on the open interval (x_0 - h, x_0 + h) and has an absolute maximum at x_0. Also, $f'(x_0) exists, following from Rolle's theorem.
The first derivative test
Example
Find the local and absolute extreme values of f(x) = x^4 - 2x^2 -3 on the interval [-2,2].
f'(x) = 4x^3 - 4x = 4x(x^2 - 1) = 4x(x - 1)(x + 1)x |
-2 |
-1 |
0 |
1 |
2 |
|---|---|---|---|---|---|
f' |
- 0 + | + 0 - | - 0 + | ||
f |
max | min | max | min | max |
| EP | CP | CP | CP | EP |