Past exam of the mathematical methods II course of the University of Oxford / 2013 resit / 11 / a / ii / 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 conditionsThe 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.