I need to minimise f(x,y)=2x+xy+3y s.t. x^2+y>=3 , x+0.5>=0, y>=0.

I get my lagrangian,

L(x,y,lambda,mu) = -lambda*x^2 – lambda*y + 3*lambda – 0.5*mu – mu*x + 2*x + xy + 3*y.

I use the Kuhn Tucker conditions for finding a min:

Lx>=0,x>=0,x*Lx=0
Ly>=0,y>=0,y*Ly=0

Llambda<=0,lambda>=0,lambda

*Llambda=0*

Lmu<=0,mu>=0,muLmu=0

Lmu<=0,mu>=0,mu

I get 3 points:

((x = 0., y = 3., lambda = 3., mu = 0.), (x = 0.3333333333, y = 2.888888889, lambda = 3., mu = 0.), (x = 1.732050808, y = 0., lambda = 0.5773502692, mu = 0.))

My question is, the constraint x+0.5>=0 is non-binding because of the KT conditions require x in the positive orthant, but surely I can not just completely ignore this constraint. Further, the latter of my 3 answers yields the lowest value of f(x,y). Is this my minimum? How do I distinguish between the points?

Many thanks