Past exam of the mathematical methods II course of the University of Oxford / 2015 resit / 5 / i / Solution

For a twice continuously differentiable function $F$, first check the stationary-point condition $F_x=F_y=0$. Write its Hessian matrix as $H=\left(\begin{array}{cc}A& B\\ B& C\end{array}\right)$, with $A=F_{xx}$, $B=F_{xy}$ and $C=F_{yy}$. The second-derivative test uses $D=AC-B^2$: if $D>0$ and $A>0$, the point is a strict local minimum; if $D>0$ and $A<0$, it is a strict local maximum; if $D<0$, it is a saddle point. If $D=0$, the quadratic test is inconclusive and higher-order terms or a direct argument are needed.

For $F=e^{y^2-x^2}$, the first derivatives vanish at the origin andThus $\boxed{(0,0)\text{ is a saddle point, with }F(0,0)=1}$. Directly, $F(x,0)=e^{-x^2}<1$ for nearby nonzero $x$, while $F(0,y)=e^{y^2}>1$ for nearby nonzero $y$. The presence of both increasing and decreasing directions confirms the Hessian matrix classification.