2 A surface has equation \(z = 2 \left( x ^ { 3 } + y ^ { 3 } \right) + 3 \left( x ^ { 2 } + y ^ { 2 } \right) + 12 x y\).
- For a point on the surface at which \(\frac { \partial z } { \partial x } = \frac { \partial z } { \partial y }\), show that either \(y = x\) or \(y = 1 - x\).
- Show that there are exactly two stationary points on the surface, and find their coordinates.
- The point \(\mathrm { P } \left( \frac { 1 } { 2 } , \frac { 1 } { 2 } , 5 \right)\) is on the surface, and \(\mathrm { Q } \left( \frac { 1 } { 2 } + h , \frac { 1 } { 2 } + h , 5 + w \right)\) is a point on the surface close to P . Find an approximate expression for \(h\) in terms of \(w\).
- Find the four points on the surface at which the normal line is parallel to the vector \(24 \mathbf { i } + 24 \mathbf { j } - \mathbf { k }\).