The question asks:
The plane $x+y+2z\:=2$ intersects the paraboloid $z\:=x^{2\:}+y^2$. Find the points on this ellipse that are nearest to and farthest to the origin.
So this is a problem on a sample final for an upcoming exam. I can tell that it's a more sophisticated problem designed to test several concepts in my multivariable class. I feel comfortable with more straightforward Lagrange problems, especially when the constraint(s) are explicitly defined, but this sort of problem is a little tough for me to wrap my head around. What exactly is the objective function (and what are the constraint(s))?
I have a solution written by a TA for this particular question. It states that there are two constraints, and that the objective function is $x^2 + y^2 + z^2$, which I believe is used to describe the ellipse. Could anyone give me tips for identifying (or discovering) what the objective function of this kind of problem is? My first thought would be that the plane function is being constrained by the paraboloid it's intersecting, but I guess I need to think about these kinds of problems in terms of what we are maximizing and minimizing.