2 A surface has equation \(z = x y ^ { 2 } - 4 x ^ { 2 } y - 2 x ^ { 3 } + 27 x ^ { 2 } - 36 x + 20\).
- Find \(\frac { \partial z } { \partial x }\) and \(\frac { \partial z } { \partial y }\).
- Find the coordinates of the four stationary points on the surface, showing that one of them is \(( 2,4,8 )\).
- Sketch, on separate diagrams, the sections of the surface defined by \(x = 2\) and by \(y = 4\). Indicate the point \(( 2,4,8 )\) on these sections, and deduce that it is neither a maximum nor a minimum.
- Show that there are just two points on the surface where the normal line is parallel to the vector \(36 \mathbf { i } + \mathbf { k }\), and find the coordinates of these points.