Proving a local minimum is a global minimum.

Let $f(x,y)=xy+\frac{50}{x}+\frac{20}{y}$, Find the global minimum / maximum of the function for $x>0,y>0$

Clearly the function has no global maximum since $f$ is not bounded. I have found that the point $(5,2)$ is a local minimum of $f$. It seems pretty obvious that this point is a global minimum, but I'm struggling with a formal proof.

Let $f(x,y)=xy+\frac{50}{x}+\frac{20}{y}$, Find the global minimum / maximum of the function for $x>0,y>0$

Clearly the function has no global maximum since $f$ is not bounded. I have found that the point $(5,2)$ is a local minimum of $f$. It seems pretty obvious that this point is a global minimum, but I'm struggling with a formal proof.