Thanks @MTyson! This is actually closer to what I had in mind.

Ha! Good point. Now I feel silly. I'll see if there is a different way of phrasing the question...

@DummieVariable Yes, if you have the conditions to perform those operations, that's one way to go.

If you're maximizing over any function $f$, you can just solve the problem separately for each $x$, and it reduces to the standard problem where you only have an integral over $y$. Calculus of variations becomes necessary if you have some constraint on $f$ etc.

@Dirk You're right, I missed that at first.