2023 HCI Promo Q11

The diagram below shows the curve \(C\) with equation \({{x}^{2}}-xy+{{y}^{3}}=16\). \(N\) is a fixed point where \(C\) intersects the positive \(x\)-axis and \(M\) is a variable point \(\left( x,y \right)\) on \(C\) where \(y>0\).

1. Show that \(\left( 3{{y}^{2}}-x \right)\frac{\mathrm{d}y}{\mathrm{d}x}-y+2x=0\).[2]
2. Show that \(\left( 3{{y}^{2}}-x \right)\frac{{{\mathrm{d}}^{2}}y}{\mathrm{d}{{x}^{2}}}+6y{{\left( \frac{\mathrm{d}y}{\mathrm{d}x} \right)}^{2}}-2\frac{\mathrm{d}y}{\mathrm{d}x}+2=0\).[2]
3. Show that \(A\), the area of triangle \(OMN\), is given by \(2y\).[2]
4. Hence find the value of \(x\) for which \(A\) has a stationary value. Determine the nature of this stationary value.[5]