I want to find pairs of two functions on two variables $f_1(x,y)$ and $f_2(x,y)$, such that each one can be minimized on one variable ($f_1$ needs to have a minimum on the variable $x$ and $f_2$ needs to have a minimum on $y$) but in such a way that their gradients are bounded, and also their graphs are bounded, and such that the minima of each happen in distinct points $(\bar{x},\bar{y})$.
For instance, if I take $f_1(x,y) = (x-y)^2$, the minimum happens when $x=y$, and if I take $f_2(x,y) = (x-(y-1))^2$, the minimum happens when $x=y-1$. The minima don't have intersection, but the functions and gradient are not bounded. I tried doing something involving sine and cosine but the problem is that there too many minima, I'd rather them have only one, or at least a global minimum...
Are there any examples satisfying all the conditions?