So I'm a bit confused with the notation of $\max_x f(x,y)$.
I am aware that it is a function of $y$, and how to calculate it is to set $y$ to a certain value and find the value of $x$ where the $f(x,y)$ is at maxima. (If I am wrong please correct me.)
However, does this $\max_x f(x,y)$ refers a function itself or just a single point??
In terms of function, I mean as in $g(y) = \max_x f(x,y)$
Or does it mean that in the end, out of all those $x$ values that I get for each $y$, I have to pick "only one" where it represents the point where the function is at the maxima.
I'm trying to understand this equation: $$\max_{a,b} f(a)g(a,b) = \max_a [f(a) \max_b g(a,b)] .$$ But I dont know what the equation represents.