Suppose $U \in \mathbb R^n$ is an open simply connected set and $f: U \to \mathbb R$ is a real valued $C^{\infty}$ function. I am wondering whether the following is possible: $f$ has more than $1$ local minimizers, say $x_1, x_2 \in U$ but does not have any other saddle points or local maximizers.
I believe if $n=1$ this cannot happen but not sure whether things change in higher dimensions.