I think there is some thing wrong in this exercise which I met by chance in a book of Calculus and analysis while looking for rigorous definition for angle , it says(adsbygoogle = window.adsbygoogle || []).push({});

let f : (a,b) → ℝ be a differentiable function , suppose that f' is bounded , and that f has a root r in (a,b) . suppose that for x ≠ r , J_{x}denote the open interval between x and r , where if f(x) > 0 then f is convex on J_{x}, and if f(x) < 0 then f is concave on J_{x}.Then prove that for any x_{0}in (a,b) the newton sequence converges to a root where x_{0}is it initial point .

The problem here is that we can take the initial point x_{0}where f'(x_{0})=0 .for the simplicity consider f(x) := x^3 for all x in (-4,4)

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# I think there is some thing wrong

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