Finding extrema (minima and maxima) of a function

See attached file for reference, please (I translated it for you, hope you understand everything).

I'm not sure what algorithm to use to find extrema over a closed set.
Any help for any subsection (1-3) is much appreciated, thanks!

Whoops, wrong number of dimensions there. Hang on.

Let me just make it clear first, that I know how to find extrema of a function of multiple variables (private derivatives etc). But my problem here is that closed set.

Also, for the first subsection the answer that I deduce is that it has no extrema... and I think I might be wrong.

Perhaps it's simplistic, but I would evaluate along all the boundaries, perhaps with a parametrization, and then use Calc I techniques to find the extrema along the boundaries. Compare those extrema with the ones in the interior of your region, and you should have it.