11 A cubic curve has equation \(y = x ^ { 3 } - 3 x ^ { 2 } + 1\).
- Use calculus to find the coordinates of the turning points on this curve. Determine the nature of these turning points.
- Show that the tangent to the curve at the point where \(x = - 1\) has gradient 9 .
Find the coordinates of the other point, \(P\), on the curve at which the tangent has gradient 9 and find the equation of the normal to the curve at P .
Show that the area of the triangle bounded by the normal at P and the \(x\) - and \(y\)-axes is 8 square units.