Suppose I have a smooth function of two variables, $f(x,y)$ defined on $U \subset \mathbb{R}^{2}$. I also impose a simple condition, say that $y<x/2$, for all $x$ and $y$.
I would like to investigate simple properties of $f(x,y)$ subject to this condition. Precisely, I suspect that $f(x,y)>0$ on a certain subset of $U$. How could I prove / disprove this, in general terms?
To me, it looks like a setup from some optimisation theory, but I am not finding min/max values.
I would be grateful if someone could point out a general "prescription", it it exists, to approach such a problem, or direct me to relevant part of mathematics that deals with such topics? Or is this something that really has to be dealt on a case-by-case basis?