Past exam of the year 1 of the computer science course of the University of Oxford / 2019 / Discrete mathematics, continuous mathematics, and probability resit / 4 / c / Solution

Let $h(x,y)=2y^3+x{y}^2-y^2-4x$, so $g=e^h>0$. Its gradient is $\nabla g=g\nabla h$, andThus a stationary point must have $y=\pm 2$ and $x=1-3y$, giving $(-5,2)$ and $(7,-2)$. To classify them, useAt either stationary point, $\nabla h=0$ and $12y+2x-2=6y$. Hence $\det H_g=g^2(-4y^2)=-16g^2<0$. The Hessian has one positive and one negative eigenvalue, so both are saddle points. There are no other stationary points.