Show that global minima exists or not

Question

My function is defined $f(x,y) = 2x^3 + 3y^2+3x^2y-24y$. I found the critical points $(x,y) = (0,4), (-2,2)$ and $(4,-4)$. I also showed that $(0,4)$ is the local minimum by showing that the Hessian at that point is positive definite. However, in order to say anything conclusive about the global minimizer, I need to say something about the function properties. As I was analyzing the function, I observed that if I take $x=0$, and let y wander off, then it seems like the local minimum I achieve could be the global minimum. However, if I take $y=0$, the function becomes $f(x) = 2x^3$ which does not have a global minima but only a saddle point at $x = 0$. I was wondering what the formal way of making such an argument would be?

Answer 1

Use the definition of a saddle point to prove that you have a global minimum. Else, use the definition of a global minimum to check whether you have a global minimum.