 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/ib/paper-2/7h/a/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

Use multipliers

λ, μ \geq 0

for

x_{1} + x_{2} - c \leq 0, x_{2} - d \leq 0.

At an interior point in the nonnegative quadrant, the Karush-Kuhn-Tucker conditions are

lo g x_{1} + 1 + λ = 0,

- 1 + λ + \frac{μ}{2 x _{2}} = 0,

with complementary slackness for the two constraints.

For

c = 3/ e^{2}

and

d = 2/ e

, the square-root constraint is inactive at the answer, so

μ = 0

. The second stationarity equation gives

λ = 1

, and the first gives

x_{1} = e^{- 2}

. The sum constraint is active, hence

x_{1} = \frac{1}{e ^{2}}, x_{2} = \frac{2}{e ^{2}} .

Indeed,

x_{2} = 2 / e < 2/ e

. The minimum value is

- \frac{4}{e ^{2}} .

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