# Method of Lagrange multipliers for multiple constraints

I'm having trouble with a past exam question regarding the use of Lagrange multipliers for multiple constraints. The question is:

Using the method of Lagrange multipliers for multiple constraints, find the absolute maximum and minimum of $$f(x,y,z) = xy + 2z$$ On the intersection of $$x + y + z =0$$ And $$x^2 + y^2 + z^2 = 24$$ I'm kinda unsure where to start, I haven't really come across many of these types of questions.

Any help would be appreciated.

You just need to consider $$F=xy+2z+\lambda(x+y+z)+\mu(x^2+y^2+z^2-24)$$ Compute $F'_x,F'_y,F'_z,F'_\lambda,F'_\mu$ and set them equal to $0$.
In this particular case where you have one linear constraint, you could eliminate $z$ from it $(z=-x-y)$ and the problem would become $$F=xy-2(x+y)+\lambda(x^2+xy+y^2-12)$$