I'm having a bit of trouble understanding what the minimum of the maximum function means.

For example, let's consider fixed $a, b$, and we are looking to find: $$\min\_{0\leq r\leq 1} \max\left(\frac{f(a)}{r},\frac{f(b)}{1-r}\right)$$

In this case, $f(a)$ and $f(b)$ are just values since we have fixed $a, b$. So we first want to find the maximum first, but how can we do that without knowing the value of $r$? What exactly does the above expression mean?