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 ?
Best Answer
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 $\mathring S$.