Past exam of the year 1 of the computer science course of the University of Oxford / 2017 / Discrete mathematics, continuous mathematics and probability / 4 / a / Solution

The gradient equations areThus $x=3$ gives $y=2$, while $x=-3$ gives $y=8$. The Hessian matrix isAt both critical points the determinant is $-36<0$, so the Hessian is indefinite and the second-derivative test classifies both as saddle points. Direct substitution givesThe factorization $f+30=(x^2-9)(x+y-5)$ also shows the crossing level-set branches through each saddle.