Past exam of the mathematics course of the University of Cambridge / 2024 / ib / Paper 2 / 7H / a / Solution

Use multipliers $\lambda ,\mu \ge 0$ forAt an interior point in the nonnegative quadrant, the Karush-Kuhn-Tucker conditions arewith 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 $\mu =0$. The second stationarity equation gives $\lambda =1$, and the first gives $x_1=e^{-2}$. The sum constraint is active, henceIndeed, $\sqrt{x_2}=\sqrt{2}/e<2/e$. The minimum value is