= Monotone Newton convergence for a strictly convex function
{title2=$e_{n+1}/e_n^2\to f''(r)/(2f'(r))$}
Let $f\in C^2([a,b])$ have $f',f''>0$ and $f(a)<0<f(b)$. Its unique root $r$ lies in $(a,b)$. Starting <Newton root-finding iteration> at $b$, strict convexity puts the tangent root strictly between $r$ and the current iterate, so the iterates decrease to $r$. Taylor expansion about the current iterate gives $e_{n+1}=f''(\xi_n)e_n^2/[2f'(x_n)]$, where $e_n=x_n-r$ and $\xi_n\in(r,x_n)$. Continuity proves the displayed positive asymptotic constant and <quadratic convergence>.
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