Past exam of the year 1 of the computer science course of the University of Oxford / 2022 / Discrete mathematics, continuous mathematics, and probability / Continuous mathematics / 6 / b / Solution

The stationary point equations give $y=x$ and $x^3=x$, hence $(0,0),(1,1),(-1,-1)$. At $(0,0)$ the Hessian matrix has negative determinant $-16$, so $(0,0)$ is a saddle point. At each of $(1,1)$ and $(-1,-1)$, the Hessian matrix has positive leading minors $12$ and $32$, proving a strict local minimum. More strongly,Thus $(1,1)$ and $(-1,-1)$ are exactly the global minima, with value $3$. There are no local maxima, since any differentiable interior local maximum would be a stationary point, and all three have been classified.