Past exam of the year 1 of the computer science course of the University of Oxford / 2023 / Discrete mathematics, probability and continuous mathematics / 6 / d / Solution

Set $h=x_{n+1}-x_n=-f(x_n)/f^{\prime }(x_n)$. By Taylor's theorem,The first two terms cancel by the definition of the Newton root-finding iteration. Thuswith $\xi$ between the consecutive iterates. In particular, a nonzero step has negative new residual under the stated strict concavity.