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Mathematics-Online lexicon:

Extrema

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A function $ f$ has a global minimum at $ a \in D$ , iff

$\displaystyle f(a) \le f(x)\quad \forall x\in D . $

For a local minimum is suffices that the value $ f(a)$ be the smallest value in some local neighborhood $ (a-\delta,a+\delta)\cap D$ .

Global and local maximum is defined analogously.

\includegraphics[width=0.6\linewidth]{Beispiel_Extrema.eps}

For piecewise continuously differentiable functions on closed intervals, extrema can only appear at those points $ b \in D$ for which $ f'(b) = 0$ , at those points $ b \in D$ for which $ f$ is discontinuous, or at boundary points. Their type can be determined by examining higher derivatives as well as by comparing various values of the function.

(Authors: Höllig/Hörner/Abele)

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  automatically generated 4/ 8/2008