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 \([-2,2]\).
| \(x\) | \(-2\) | \(-1\) | \(0\) | \(1\) | \(2\) |
|---|---|---|---|---|---|
| \(f'\) | - 0 + | + 0 - | - 0 + | ||
| \(f\) | max | min | max | min | max |
| EP | CP | CP | CP | EP |