Let $x$ and $y$ be positive real numbers, and let $m$ be the minimum value among $x,$ $\frac{1}{y},$ and $y + \frac{1}{x}.$ Find the largest possible value of $m.$
So basically I need to find $$\max_{x,y>0}\left(\min\left\{x,\dfrac{1}{y},y+\dfrac{1}{x}\right\}\right).$$
I tried to consider some cases for $x$ and $y$, and doing the problem in each of those cases, but that didn't get me anywhere. Could anyone give me any hints on how to proceed?
Thanks!!