= Solution
At an interior <stationary point>, let $A=f_{xx}$, $B=f_{xy}$, $C=f_{yy}$ and $D=AC-B^2$. The quadratic change in the <Taylor expansion> is negative in every nonzero displacement when the <Hessian matrix> is negative definite. Applying the minimum test to $-f$ gives the sufficient nondegenerate maximum conditions
$$
\boxed{f_x=f_y=0,\qquad A<0,\qquad D>0.}
$$
The quadratic term then dominates the smaller Taylor remainder in a sufficiently small neighborhood, proving a strict local maximum. A vanishing determinant requires higher-order analysis instead.
Back to article page
