How do we find the critical point using partial derivatives? If $\frac{\partial f}{\partial x}= 0$ when $x = 1$, and $\frac{\partial f}{\partial x} = (2y-x)$
Do we have to replace $x$ by 1? to get the critical coordinate $y$?

The critical points occur when $f_{x}=f_{y}=0$.  So, necessarily, any critical point must occur when $x=1$ so that we obtain $2y-1=0$ and $y=\frac{1}{2}$ as desired.  So you are correct.
