Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 38 / 1 / b / Solution

With $g=2x-y-z-2$ and $h=x^2+y^2-5$, the maximization optimization Lagrangian isA finite unconstrained maximum requires $\lambda =1$ to cancel the coefficient of $z$. With $\mu =1$, completing the square givesThus its global maximizers have $x=-1$ and $y=2$. The equality constraint holds, and complementary slackness with $\lambda =1$ makes $g=0$, giving $z=-6$. The Lagrangian sufficiency theorem certifiesEvery feasible point has objective at most $L\le 12$, so this is a global conclusion rather than just a stationary-point calculation.