6 The surface \(S\) has equation \(z = x \sin y + \frac { y } { x }\) for \(x > 0\) and \(0 < y < \pi\).
- Determine, as a function of \(x\) and \(y\), the determinant of \(\mathbf { H }\), the Hessian matrix of \(S\).
- Given that \(S\) has just one stationary point, \(P\), use the answer to part (a) to deduce the nature of \(P\).
- The coordinates of \(P\) are \(( \alpha , \beta , \gamma )\).
Show that \(\beta\) satisfies the equation \(\beta + \tan \beta = 0\).