Given a function $f: \mathbb{R}^n \rightarrow \mathbb{R}$ which has only one critical point and it's a local minimum, for what $n$ is it a global minimum?
For a convex function with one variable a local minimum is always global.
For functions with two variables, it's not true. There are many counterexamples:
$f(x,y) = e^{3x} + y^3-3ye^x$.
Here the only solution of $f_x=3e^{3x}-3ye^x=0$, and $f_y=3y^2-3e^x=0$ is $(0,1)$ which is a local minimum by the second derivative test.
But $f(0,-3)=-17<f(0,1)=-1$
$f(x,y)=x^2+y^2(1+x)^3$ has the same property.
What about higher dimensions?
Could you help me determine the condition on $n$ for which the only local minimum is global?
Thank you.