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Extrema on Compact Sets

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Let $ f$ be a continuous real multivariate function defined on a compact set $ M$ (i.e.the set $ M$ is bounded and closed). Then $ f$ has a minimum and a maximum on $ M .$ In other words there exist a maximum point $ x_1$ such that $ f(x_1) \geq f(x) $ for all $ x \in M$ and a minimum point $ x_2$ with $ f(x_2) \leq f(x) $ for all $ x \in M .$

Note that there may exist more than one maximum point and more than one minimum point.

\includegraphics[width=0.35\linewidth]{Extremwerte_mF_bild1} \includegraphics[width=0.35\linewidth]{Extremwerte_mF_bild2}

The picture shows the graph of a function of two variables with several extrema. On the right hand side the corresponding level curves are shown.

Example:

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  automatically generated 1/23/2017