2 A surface, S , has equation \(z = 3 x ^ { 2 } + 6 x y + y ^ { 3 }\).
- Find the equation of the section where \(y = 1\) in the form \(z = \mathrm { f } ( x )\). Sketch this section.
Find in three-dimensional vector form the equation of the line of symmetry of this section.
- Show that there are two stationary points on S , at \(\mathrm { O } ( 0,0,0 )\) and at \(\mathrm { P } ( - 2,2 , - 4 )\).
- Given that the point ( \(- 2 + h , 2 + k , \lambda\) ) lies on the surface, show that
$$\lambda = - 4 + 3 ( h + k ) ^ { 2 } + k ^ { 2 } ( k + 3 ) .$$
By considering small values of \(h\) and \(k\), deduce that there is a local minimum at P .
- By considering small values of \(x\) and \(y\), show that the stationary point at O is neither a maximum nor a minimum.
- Given that \(18 x + 18 y - z = d\) is a tangent plane to S , find the two possible values of \(d\).