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Minimise $C(x,y)=11x+3y$ subject to the constraints $ g(x,y)=-3x^2-3y^2+10xy $ and $x\geq 0, y\geq 0$. I started solving using this Lagrange multiplier, but the constraint set is not compact, right? So, how can I argue whether the critical point is a maximum or a minimum or a saddle point?

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What is your constraint on g(x,y)? There should be some equality or inequality, associated with that equation.

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