Past exam of the mathematical methods II course of the University of Oxford / 2013 resit / 11 / c / i / Solution

On $y=0$, the restricted function is $g(x)=x^4$, so $g^{\prime }(x)=4x^3$ changes from negative to positive at zero. HenceThe restricted second derivative vanishes there, but the quartic term determines the minimum. This does not contradict the two-dimensional saddle: values decrease along other directions, including the diagonal near the origin.