I understand that for concave and convex functions in $\Re^n$, the extremum will occur at a stationary point and that concavity/convexity guarantees the existence of such a point whenever there is an interior solution; but I don't understand how we can confidently exclude a case where a function does not have any stationary points (e.g. A case where the function is stationary at one or more coordinate(s) in some neighborhood of its domain, but never stationary at all coordinates) but still has an interior solution on some subset of $\Re^n$?
EDIT: I've worked on this some more, and it seems like a function as described above would not have an interior solution, can anyone confirm this or direct me to a formal proof?