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?