Minimization Generalization

Geometry Level 3

I've seen a couple of variants of problem where we are to minimize a function which is a sum of square roots. Let's generalize a solution!

Let \( f(x,y)=\sqrt{(x+a)^2+(y+b)^2}+\sqrt{(x-c)^2+(y-d)^2} \) for fixed \(a\), \(b\), \(c\) and \(d\).

The function \(f\) has a global minimum which can be expressed as \(\sqrt{p(a,b,c,d)}\) where \(p\) is a polynomial over \(\mathbb{N}\). Find the sum of the coefficients of \(p\).

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