Solution
= Solution
Set $h=x_{n+1}-x_n=-f(x_n)/f'(x_n)$. By <Taylor's theorem>,
$$
f(x_{n+1})=f(x_n)+hf'(x_n)+\frac{h^2}{2}f''(\xi).
$$
The first two terms cancel by the definition of the <Newton root-finding iteration>. Thus
$$
\boxed{f(x_{n+1})=\frac{(x_{n+1}-x_n)^2}{2}f''(\xi),}
$$
with $\xi$ between the consecutive iterates. In particular, a nonzero step has negative new residual under the stated strict <concavity>.