How to ensure extreme? -- using Extreme Value Theorem

Question

I think it's a simple question. How can I ensure the existence of an extreme (maximum or minimum) using the Extreme Value Theorem / Weierstrass theorem?

For example, this multivariate case: $$ F(x,y)=\frac{y^3}{3}-\frac{x^3}{3}+x^2 y - y^2 -3y. $$

Answer 1

In general, you simply can't. Essentially there is no general tool for locating local extrema. Weierstrass' Theorem applies only to continuous functions defined on compact sets, but is totally useless in other cases.

Think of the function $f(x)=x+\sin x$, which possesses a sequence of local extrema. However, it is very hard to predict their existence without solving $f'(x)=0$ and checking the nature of each zero of the derivative. As you can imagine, the situation gets even harder for functions of several variable, since you can approach a point along infinitely many directions.