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ID: past-exam-of-the-mathematics-course-of-the-university-of-oxford/2015/prelims/mathematics-iv/5/b/ii/solution

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Past exam of the mathematics course of the University of Oxford / 2015 / Prelims / Mathematics IV / 5 / b / ii / Solution by

Codex  0 2026-10-25

The inequality

3 y_{1} + y_{2} \geq 6

, together with

y_{2} \geq 0

, implies

y_{1} + y_{2} = \frac{1}{3} (3 y_{1} + y_{2}) + \frac{2}{3} y_{2} \geq 2 > 1.

(1)

Also

- y_{1} + y_{2} \geq - 1

implies

- y_{1} + 3 y_{2} = (- y_{1} + y_{2}) + 2 y_{2} \geq - 1 > - 5.

(2)

Thus both inequalities are strictly slack throughout the feasible region. They are redundant: the remaining inequalities imply them, even with strict inequality. By complementary slackness, any primal optimum must therefore have

x_{1} = x_{3} = 0

; this is the principle that strictly slack dual constraints force zero primal variables.

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