# Unique index 0 critical point if all critical points are of index 0?

Let $$f:\mathbb{R}^n\to\mathbb{R}$$ be a Morse function whose all critical points are of index 0. Is it true that $$f$$ has a unique critical point if it has at least one critical point?

Since $$\mathbb{R}^n$$ is not compact, the relation between the number of critical points of different indices and the Euler characteristic does not apply here. So I have no idea about this question currently. Thank you.

This is false for compactness reasons. You can not do this in $$\mathbb{R}$$, but there exists a function on $$\mathbb{R}^2$$ with two local minima and no other critical points.
It is a bit hard to write an explicit formula, but consider the graph of the function $$f(x,y)=(x-1)^2(x+1)^2+y^2$$. This has two minima at $$(\pm 1,0)$$ and one saddle at $$(0,0)$$. Imagine dragging the saddle point to $$(0,\infty)$$.
Another way of seeing that you can do this is to Look at the function on $$X=R^2\setminus [-1/2,1/2]\times [1/2,\infty)$$, where it has only the two minimia. Convince yourself that $$X$$ is diffeomorphic to $$R^2$$.