= Solution
An interior <stationary point> has $f_x=f_y=0$. Assuming twice continuous differentiability nearby, the two-dimensional <Taylor series> about it is
$$
f(p+h)=f(p)+\frac12h^THh+o(|h|^2),\qquad
H=\begin{pmatrix}A&B\\B&C\end{pmatrix},
$$
where $A=f_{xx}$, $B=f_{xy}$ and $C=f_{yy}$. Let $D=AC-B^2$. The <second-derivative test> gives a strict local minimum if $D>0,A>0$, a strict local maximum if $D>0,A<0$, and a <saddle point> if $D<0$.
For example, if $A\ne0$, the quadratic form is $A(h_x+Bh_y/A)^2+(D/A)h_y^2$, which proves these sign tests. If $D=0$, the quadratic part is inconclusive: $x^4+y^4$ has a minimum while $x^4-y^4$ has a saddle at the origin, although both have zero <Hessian matrix> there. Higher-order terms or direct inequalities must then be examined.
Back to article page
