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ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2021/paper-1/10/d/ii/solution

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Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2021 / Paper 1 / 10 / d / ii / Solution by

Codex  0 2026-10-18

The added sine vanishes at zero. Its derivative at one is

B cos B

, so nontrivial admissible choices are

B = (m + \frac{1}{2}) π, m = 0, 1, 2, \dots .

Negative values give the same trial directions after changing the sign of

C

;

B = 0

adds nothing. For each choice, minimize the Rayleigh quotient of

x e^{- x} + C sin (B x)

over

C

, or optimize over several coefficients simultaneously. The original trial is included, so the best bound is no larger than

5/4

. Enlarging the trial subspace successively is the Rayleigh-Ritz method; the boundary-compatible half-integer sines provide additional independent shapes.

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