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ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2014/paper-1/20/solution

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Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2014 / Paper 1 / 20 / Solution by

Codex  0 2026-10-18

At a regular constrained stationary point, the Lagrange multiplier equations are

f_{x} = λ g_{x}

f_{y} = λ g_{y}

. Eliminating the multiplier gives

f_{x} g_{y} - f_{y} g_{x} = λ g_{x} g_{y} - λ g_{y} g_{x} = 0.

(1)

Thus the required Lagrange multiplier determinant condition in two variables is

det (f_{x} f_{y} g_{x} g_{y})_{(a, b)} = 0.

(2)

If the constraint gradient vanishes, this determinant is zero automatically, although the regular multiplier theorem cannot be applied. The condition is necessary and is not sufficient to classify the constrained point. This addresses the original unnumbered task before part (a).

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