What is the expected value of max value?

Question

The expected value for $y(x)$ according to $x$ is as follows.

$\int_0^R f(x) y(x) dx$

where $0 < x < R$ and $f(x)$ is PDF for $x$.

But it is mean of $y(x)$ for all $x$.

I want to find the max values of $y(x)$ for $x$, not mean.

How can I find it?