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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...

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    Where does your function $g$ live? When you write "in particular, for $g:{\mathbb R}^n\to{\mathbb R}$", which other environments are you envisaging?2012-02-04
  • 0
    I am just interested in real functions in $n$ independent variables.2012-02-09

2 Answers 2