= Solution
At a <stationary point>, the linear terms vanish, so $\Delta f\simeq\tfrac12(Ah^2+2Bhk+Ck^2)$, with $A=f_{xx}$, $B=f_{xy}$, $C=f_{yy}$. Let $D=AC-B^2$ be the <Hessian determinant>. For $A\ne0$, completing the square gives
$$
2\Delta f\simeq A\left(h+\frac BA k\right)^2+\frac DA k^2.
$$
The <second-derivative test> therefore yields
$$
\boxed{D>0,\ A>0:\text{ strict local minimum};\quad
D>0,\ A<0:\text{ strict local maximum};\quad
D<0:\text{ saddle point}.}
$$
When $D<0$ the quadratic form is indefinite, including the case $A=0$, $B\ne0$. When $D=0$, the quadratic approximation is degenerate and the test is inconclusive: higher-order terms may produce a minimum, maximum or saddle. For nondegenerate cases the quadratic term dominates the Taylor remainder sufficiently close to the point.
Back to article page
