# Show that global minima exists or not

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?

• If you put dollar signs around your math expressions, you'll find they look ever so much nicer. You'll also see that you'll be able to dispense with the asterisks to denote multiplication. (There's more to learn about TeX, but this is enough for this question, and will get you off to a good start.) – Barry Cipra Feb 7 at 20:21
• @BarryCipra thank you for the suggestion. I will implement it in the future. – Yukti Kathuria Feb 7 at 20:25
• no upper bound and no lower bound. This is a little more detailed than just saying no global minimum point; for example, $e^x$ has no global minimum point but it does have a lower bound, namely $0$ – Will Jagy Feb 7 at 20:32