For starters, picture the problem in 3D. You have an ellipsoid with cardinal points . Hopefully you can visualize that the local minima and maxima of the distance function fall exactly on these points.

Secondly, the equations you set up are intended as a system of four equations and four unknowns:

OR

OR

OR

The only logical solutions to this set of relations is

The values of lambda (for your purpose) are "scrap" variables. The real purpose of the system of equations is to solve for point(s) . However, if you look carefully, the value is precisely the square of the distance at its corresponding point . So to answer your question, represents the output of the function you are minimizing, for reasons beyond the scope of this question.