2 A surface \(S\) is defined by \(z = 4 x ^ { 2 } + 4 y ^ { 2 } - 4 x + 8 y + 11\).
- Show that the point \(\mathrm { P } ( 0.5 , - 1,6 )\) is the only stationary point on \(S\).
- On the axes in the Printed Answer Booklet, draw a sketch of the contour of the surface corresponding to \(z = 42\).
- By using the sketch in part (b)(i), deduce that P must be a minimum point on \(S\).
- In the section of \(S\) corresponding to \(y = c\), the minimum value of \(z\) occurs at the point where \(x = a\) and \(z = 22\).
Find all possible values of \(a\) and \(c\).