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Would be a great help if someone could give me some good links on this topic or some methods generally.

If the domain is compact; where are the points that could possibly maximize/minimize the function?

Is it either points that satisfy the equation [itex]\nabla[/itex][itex]f = 0[/itex] and points on the boundary?

In one problem I did, the point that maximized the function didn't satisfy [itex]\nabla[/itex][itex]f = 0[/itex], how come?

How do I examine the boundary? if the domain is defined by an inequality and the equality corresponds to the boundary, do I just solve for either variable and plug into the original equation? What if it's a three variable function?

If the domain isn't compact, and both x and y go from 0 to infinity, what do I do then?