Past exam of the mathematics course of the University of Cambridge / 2022 / ib / Paper 2 / 7H / Solution

The Lagrange sufficiency theorem says that for a convex differentiable objective and convex differentiable inequality constraints, any feasible point satisfying the Karush-Kuhn-Tucker conditions with nonnegative multipliers is a global minimizer.

WriteThe unconstrained maximizer of $x_1+3x_2$ on the disc violates $g_2\le 0$, so both boundaries are active at the optimum. Their intersections areand $(3,4)$ gives the smaller objective.

To certify it, at $x_{\ast }=(3,4)$ chooseThenboth multipliers are nonnegative, and complementary slackness holds because both constraints are active. The sufficiency theorem therefore gives