The curve \(C\) has equation \(x^{3}+y^{3}=3 x y\), for \(x\gt 0\) and \(y\gt 0\). Find a relationship between \(x\) and \(y\) when \(\frac{\mathrm{d} y}{\mathrm{~d} x}=0\).

Find the exact coordinates of the turning point of \(C\), and determine the nature of this turning point.