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Homework Statement
What I don't understand is why you can maximize the distances squared  d^{2}. Isn't d^{2} different from d? I don't see how they can get you the same value.
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They don't have the same values (unless they happen to be 0 or 1), but they are maximized or minimized at the same points (x,y,z). Think about it: how could it be otherwise?Homework Statement
What I don't understand is why you can maximize the distances squared  d^{2}. Isn't d^{2} different from d? I don't see how they can get you the same value.
As HallsofIvy has pointed out, the stationary points are the same in both problems. However, if the original f can take both negative and positive values, a min in the squared problem can be a saddle point in the original problem (eg: f(x) = x^3 has a saddle point at x = 0 but g(x) = f(x)^2 = x^6 has a global minimum at x = 0) and a min in the original problem can be a max in the squared problem, etc. None of these issues arise if the original f is >= 0, as it is in the distance problem you cite.They don't have the same values (unless they happen to be 0 or 1), but they are maximized or minimized at the same points (x,y,z). Think about it: how could it be otherwise?
RGV