Consider a function $g$ with the following properties.
- It is smooth.
- $g > 0$.
- $g \to 0$ at infinity.
- It has at least two critical points.
- There are finitely many critical points.
- Each critical point is isolated.
Thanks to the answer below, I am going to add one additional restriction on $g$.
- $g$ is a rational function.
I am adding yet another condition after seeing an edit below.
- Each critical point of $g$ is non-degenerate; that is, if $x$ is a critical point then $\det g''(x) \neq 0$.
In the example below, the critical point that is not a saddle has a zero eigenvalue and hence the determinant is zero.
Notice at least one of the critical points has to be a local max.
The question is: does $g$ have a saddle point?
In particular, for $g \colon \mathbb{R}^n \to \mathbb{R}$, does $g$ have a critical point of index $n-1$?
If there is a reference you can point me to that would be terrific. I believe a variant of the Mountain Pass Theorem may work...