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  • $\begingroup$ The points of maximum error satisfy $f'(x) = p'(x)$. Does that help? $\endgroup$ Commented May 15, 2021 at 18:16
  • $\begingroup$ Right so that's a rephrasing of where the critical points of the error function are. $f(x) - p(x)$ is the error function (sans absolute value) and $f'(x) - p'(x) = 0$ is where the critical points are and where $f(x) - p(x)$ I wasn't clear about it but that's what I was referring to with "at least one critical point" method. $\endgroup$ Commented May 15, 2021 at 18:33
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    $\begingroup$ You can find a critical point between any two solutions of $f(x) = p(x)$. $\endgroup$ Commented May 15, 2021 at 18:34
  • $\begingroup$ Ah ok, so E in the Remez algorithm is the error of approximation at the sample points and and it alternates so we can find a root between those points giving us N+1 roots...now that you have N+1 roots you can find N roots of f'(x) - p'(x) between the roots of f(x) - p(x) and those are your points of maximum error. You can add in the end points as freebie critical points to get N+2 new samples! $\endgroup$ Commented May 15, 2021 at 18:56
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    $\begingroup$ You’ll have to detect it somehow. The algorithm will seem to fail. $\endgroup$ Commented May 16, 2021 at 4:58