= Monotonicity of polynomial critical points in their roots
{title2=$\frac{\partial\xi}{\partial a_j}=\frac{(\xi-a_j)^{-2}}{\sum_i(\xi-a_i)^{-2}}>0$}
For a <monic polynomial> $p(x)=\prod_i(x-a_i)$ with distinct ordered real <roots of a polynomial>, each <critical point> $\xi$ lies between two consecutive roots by <Rolle's theorem>. Its <logarithmic derivative> gives $\sum_i(\xi-a_i)^{-1}=0$. <Implicit differentiation> gives the displayed formula, so moving any root to the right moves every ordered <critical point> to the right, while the roots remain distinct. Iterating this statement for <derivatives> preserves componentwise ordering of their roots. Weak inequalities extend by <continuity> to repeated roots.
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