Exercise 1.7.25

The corresponding matrix $S=\left[\begin{array}{cc}\hfill 4\hfill & \hfill 6\hfill \\ \hfill 6\hfill & \hfill c\hfill \end{array}\right]$. To have a bowl, we want the matrix $S$ to be positive definite, i.e. $4c-36>0$, $c>9$. To have a saddle point, we want to have both positive and negative eigenvalues, $c<9$.

When $c=9$, we have $z=4x^2+12\mathit{xy}+9y^2={(2x+3y)}^2$. We have lines correspond to each fixed $z$. So the graph is a ‘trough’ which stays zero along the line $2x+3y=0$.