If the critical points corresponding to the global min/max of a function ##f:\mathbb{R}^2\rightarrow\mathbb{R}## lie in a subset ##A## of ##\mathbb{R}^2##, then the global min/max of ##f## in ##A## correspond to the global min/max of ##f##.(adsbygoogle = window.adsbygoogle || []).push({});

If the global min/max of ##f## lie outside of ##A##, then either the global min/max of ##f## in ##A## are in the boarders of ##A## or are the local min/max of ##f## that lie in ##A##. (There is no way that the global min/max of ##f## in ##A## is not a critical point of ##f## or in the boarders of ##A##, right?)

And finally if the global min/max of ##f## in ##A## lie in the boarders of ##A##, then either the subset ##A## is closed in which case the global min/max corresponds to those points in the boarders, or the subset ##A## is open in which case (say that the point in the boarder corresponding to the max is the point ##(a,b)##) the global max would then be:

[tex] \lim_{(x,y)\rightarrow{}(a,b)}f(x,y). [/tex]

Is that right? Did I miss something? Are there cases I left out?

By the way, I think that what I mean when I say the global min max of ##f:\mathbb{R}^2\rightarrow\mathbb{R}## in ##A## is the global min/max of the restricted function ##f:A\rightarrow\mathbb{R}## although I am not entirely sure.

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# Min/max of functions restricted to subsets of their domains

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