 Solution

ID: past-exam-of-the-mathematical-methods-ii-course-of-the-university-of-oxford/2013-resit/11/a/ii/solution

A = f_{xx}

B = f_{x y}

C = f_{yy}

and

D = A C - 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

f_{x} = f_{y} = 0, A < 0, 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.

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