N2012 P1 Q8

The curve \(C\) has equation \(x-y={{\left( x+y \right)}^{2}}\). It is given that \(C\) has only one turning point.

1. Show that \(1+\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{2}{2x+2y+1}\).[4]
2. Hence, or otherwise, show that \(\frac{{{\mathrm{d}}^{2}}y}{\mathrm{d}{{x}^{2}}}=-{{\left( 1+\frac{\mathrm{d}y}{\mathrm{d}x} \right)}^{3}}\).[3]
3. Hence state, with one reason, whether the turning point is a maximum or a minimum.[2]