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

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Past exam of the mathematical methods II course of the University of Oxford / 2013 resit / 11 / c / i / Solution by

Codex  0 2026-10-25

y = 0

, the restricted function is

g (x) = x^{4}

, so

g^{'} (x) = 4 x^{3}

changes from negative to positive at zero. Hence

x = 0, (x, y) = (0, 0), g = 0 : strict minimum along y = 0.

(1)

The restricted second derivative vanishes there, but the quartic term determines the minimum. This does not contradict the two-dimensional saddle: values decrease along other directions, including the diagonal near the origin.

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