2 A surface has equation \(z = 3 x ^ { 2 } - 12 x y + 2 y ^ { 3 } + 60\).
- Show that the point \(\mathrm { A } ( 8,4 , - 4 )\) is a stationary point on the surface. Find the coordinates of the other stationary point, B , on this surface.
- A point P with coordinates \(( 8 + h , 4 + k , p )\) lies on the surface.
(A) Show that \(p = - 4 + 3 ( h - 2 k ) ^ { 2 } + 2 k ^ { 2 } ( 6 + k )\).
(B) Deduce that the stationary point A is a local minimum.
(C) By considering sections of the surface near to B in each of the planes \(x = 0\) and \(y = 0\), investigate the nature of the stationary point B . - The point Q with coordinates \(( 1,1,53 )\) lies on the surface.
Show that the equation of the tangent plane at Q is
$$6 x + 6 y + z = 65$$
- The tangent plane at the point R has equation \(6 x + 6 y + z = \lambda\) where \(\lambda \neq 65\).
Find the coordinates of R .