Past exam of the mathematical methods II course of the University of Oxford / 2014 / 8 / b / Solution

At a smooth interior stationary point, define $D=t_{11}{t}_{22}-t_{12}^2$. The second-derivative test gives a minimum for $D>0,t_{11}>0$, a maximum for $D>0,t_{11}<0$, and a saddle point for $D<0$; $D=0$ is inconclusive.

For the stated time function, putPositive physical speeds make all three coefficients positive. The positive Hessian of a layered travel-time functional here isThus the stationary path is a strict minimum, not merely a candidate of unspecified type. In fact the same positivity holds for all finite crossing positions, so the time function is strictly convex and the minimum is unique.