Dear Bob, 

While drinking beer yesterday several ideas came about our maximization problem and the question of root selection. You have probably solved the problem by now, but here are a few points anyhow. First, as you know, the solution is a simple saddle point of [equation], gotten by setting both partials to zero. The problem is how to select between the 3 roots which sometimes occur. I have thought a little about why the “magic” multipliers (Lagrange) work to express constraints, and will try to briefly give you the picture: If we have a function f(xi) to be maximized subject to constraints gj(xi) = cj,[?] then we can picture the constraints as limiting our region of interest to a “surface” (subspace) within the xi space. [erasure] There is in fact a whole family of surfaces which can be labeled by the value of the gj(xi) which are constant over any one surface. 

If we now choose some fixed values [lambda]1, [lambda]2     [lambda]n