2024 HCI Promo Q6

The curve \(C\) has equation \(\ln {{y}^{2}}={{x}^{2}}y\), where \(x\in \mathbb{R}\), \(-1\le y<0\).

It is given that \(C\) has only one turning point.

1. Show that \(\frac{\mathrm{d}y}{\mathrm{d}x}\left( 2-{{x}^{2}}y \right)=2x{{y}^{2}}\).[3]
2. Find the coordinates of the turning point.[2]
3. Calculate the value of \(\frac{{{\mathrm{d}}^{2}}y}{\mathrm{d}{{x}^{2}}}\) at the turning point and determine whether the turning point is a maximum or a minimum.[3]