How to check if function has global extremes? I need to determine if function:

$f(x,y)=x+2y-2\log(xy) $

has global minimum/maximum.

I've found local minimum at $(2,1)$, but that's not any proof of global minimum.
Put $x=1/y$. You get $f(x,1/x)=x+2/x$. Approach $x$ to zero from below and from above. What do you get?
Hint: $$f(x,y)=x+2y−2\log(xy) = (x-2 \log(x)) + (2y-2 \log(y))$$ so you have a separable function and can look for minima for $x$ and for $y$ individually.

So long as both are positive, it is obvious that $(2,1)$ is the only point giving a minimum and that there is no maximum.