Knowlpedia

Global extrema

A continuous real function on a compact set attains its maximum and minimum.

Global extrema: Let K⊆RnK\subseteq\mathbb R^n be nonempty and compact, and let f:K→Rf:K\to\mathbb R be . Then there exist points xmin⁡,xmax⁡∈Kx_{\min},x_{\max}\in K such that

f(xmin⁡)≤f(x)≤f(xmax⁡)for all x∈K.f(x_{\min})\le f(x)\le f(x_{\max}) \quad \text{for all } x\in K.

On a closed [a,b][a,b], this theorem ensures the existence of global maximizers and minimizers needed in many basic arguments in differential and integral calculus.

Remarks

In particular, ff has a and a on KK.