A Two variable function $$f(x,y)=(x-2)^2 +(y-2)^2 $$ The Domain of x is $[0,4]$ and Domain of $y$ is $[0,4]$.
How I can determine the supremum $$ g(x) = \sup_y f(x, y)$$ and infimum $$h(y) = \inf_xf(x, y)$$ of the function.
For $g(x)$ I guess I find derivate of $f(x,y)$ with respect to $x$ which turns out to be $2(x-2)$ and for $h(y)$ i find derivate of $f(x,y)$ with respect to which turns out to be $2(y-2)$
Notice that this is a separable problem, the $x$ and $y$ can be optimized independently.
$$g(x) =\sup_y f(x,y) = (x-2)^2 + \sup_y (y-2)^2 =(x-2)^2+4$$
$$h(y) =\inf_x f(x,y) = \inf_x(x-2)^2 + (y-2)^2 =(y-2)^2$$