The exact procedure of calculating $\max_{x} f(x,y)$?

Question

What is the exact procedure to calculate $\max_{x} f(x,y)$?

This is my opinion:

1. Choose one certain $y = y_0$, and then scan all $x$,find the max value of $f(x, y_0)$ named $f_0$;
2. Choose another certain $y = y_1$, and do the same thing, I can get $f_1$;
3. Repeat 1 and 2, I can get a function named $g(y)$, and $g(y) = \max_{x} f(x,y)$

Is that right?

Similar question: How to interpret max(min(expression))??

Answer 1

You are correct that $\max_x f(x,y)$ is a function of $y$, so for every $y$, there is some specific value that equals $\max_x f(x,y)$.

Example:

For example, taking $f(x,y)=\sin x \sin y$

will mean that $\max_x f(x,y)= \sin y$.