BALTIC-WAY 1999 Problem 4

For all positive real numbers $x$ and $y$ let

$$f(x, y)=\min \left(x, \frac{y}{x^{2}+y^{2}}\right) .$$

Show that there exist $x_{0}$ and $y_{0}$ such that $f(x, y) \leqslant f\left(x_{0}, y_{0}\right)$ for all positive $x$ and $y$, and find $f\left(x_{0}, y_{0}\right)$.