# Prove that a multivariable function doesn't have global extremes

So my question is actually this.

Say I have a function $F:\mathbb R^2\to\mathbb R$. If I find all the potential local extremes by finding the roots of the partial derivatives and I find that only one of them is an actual minimum while the rest are saddles, is it a good enough argument to just find a point that's larger than the local minimum I found and a point that's lower than it to prove that there are no global extremes?

Actually, since there are no local maximums, just finding a point that's lower than the local minimum should be enough, right?

If it isn't, can you provide an example where that isn't the case?

You're also completely correct about the local minimum. If you have a local minimum, $x^*$, then by definition, it is minimal within some neighborhood. If you find a point which produces a lower value of $F$ (which would need to come from outside this neighborhood), then the neighborhood within which $x^*$ is a minimizer must be bounded. Then $x^*$ is not minimal over all of $\mathbb{R}^2$ and $x^*$ is not a global minimizer.