2 A surface has equation \(z = 3 x ( x + y ) ^ { 3 } - 2 x ^ { 3 } + 24 x\).
- Find \(\frac { \partial z } { \partial x }\) and \(\frac { \partial z } { \partial y }\).
- Find the coordinates of the three stationary points on the surface.
- Find the equation of the normal line at the point \(\mathrm { P } ( 1 , - 2,19 )\) on the surface.
- The point \(\mathrm { Q } ( 1 + k , - 2 + h , 19 + 3 h )\) is on the surface and is close to P . Find an approximate expression for \(k\) in terms of \(h\).
- Show that there is only one point on the surface at which the tangent plane has an equation of the form \(27 x - z = d\). Find the coordinates of this point and the corresponding value of \(d\).