Prove that function has minimum when Hessian is zero

Question

Prove that the following function has a minimum at $(0,0)$.

$$ f(x,y) = \dfrac{1}{24}[e^{2y+2\sqrt{3} x}+e^{2y-2\sqrt{3} x} + e^{-4y}] - \dfrac{1}{8} $$

My attempt:

I tried to solve it via hessian criterion but determinant of hessian is zero so the test via hessian is inconclusive. I then tried to factor my original function and I get:

$$f(x,y) = \dfrac{1}{24}[2e^{2y} \cosh(2\sqrt{3}x) + e^{-4y}] - \dfrac{1}{8}$$

I notice that terms in the brackets are always positive, but I do not see how to conclude I have a minimum (if it is possible to conclude that from the given form of the function).

Thank you for any help.

Answer 1

It's $0$ at $(0,0)$ and by using AM-GM on everything inside the bracket $f(x,y)\geq 0$