6 A surface \(S\) is defined by \(z = \mathrm { f } ( x , y ) = 4 x ^ { 4 } + 4 y ^ { 4 } - 17 x ^ { 2 } y ^ { 2 }\).
- Show that there is only one stationary point on \(S\).
The value of \(z\) at the stationary point is denoted by \(s\).
- State the value of \(s\).
- By factorising \(\mathrm { f } ( x , y )\), sketch the contour lines of the surface for \(z = s\).
- Hence explain whether the stationary point is a maximum point, a minimum point or a saddle point.
C is a point on \(S\) with coordinates ( \(a , a , \mathrm { f } ( a , a )\) ) where \(a\) is a constant and \(a \neq 0\). \(\Pi\) is the tangent plane to \(S\) at C .
- Find the equation of \(\Pi\) in the form r.n \(= p\).
- The shortest distance from the origin to \(\Pi\) is denoted by \(d\). Show that \(\frac { d } { a } \rightarrow \frac { 3 \sqrt { 2 } } { 4 }\) as \(a \rightarrow \infty\).
- Explain whether the origin lies above or below \(\Pi\).
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