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I was watching https://www.youtube.com/watch?v=Hg38kfK5w4E&ab_channel=TheOrganicChemistryTutor about finding global max and min for a multivariable function.

He said that they occur at

at the corners of the rectangle , at some point along one of the borders, at the critical "or some other point inside this region.

Are they all true? Someone in the comment pointed out that he made a mistake?

So in general, what types of points can potentially be global maxima or minima for a multivariable function and why ?

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If $S$ is a closed and bounded subset of $\Bbb R^n$ and if $f\colon S\longrightarrow\Bbb R$ is a continuous map which is differentiable in the interior $\mathring S$ of $S$, then $f$ has a maximum and a minimum in $S$. Furthermore, the points at which the maximum and the minimum are attained must be located at the boundary $\partial S$ of $S$ or at the critical points of $f$ in $\mathring S$.

If $S$ is closed but unbounded, then $f$ can fail to have a maximum or a minimum, but it is still true that, if it has, then they are attained at the boundary $\partial S$ of $S$ or at the critical points of $f$ in $\mathring S$.

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  • $\begingroup$ Thank you Dr Santos, so he is wrong and the global extrema cannot happen in the interior of the square/region? $\endgroup$
    – CountDOOKU
    Commented Oct 14, 2021 at 10:25
  • $\begingroup$ Yes, that is wrong. Just consider $f(x,y)=x^2+y^2$ in a square centered at $(0,0)$. Clearly, the minimum is attained at $(0,0)$. $\endgroup$
    – José Carlos Santos
    Commented Oct 14, 2021 at 10:29

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