Monotone Newton convergence for a strictly convex function ($e_{n+1}/e_n^2\to f^{\prime \prime }(r)/(2f^{\prime }(r))$)

Let $f\in C^2([a,b])$ have $f^{\prime },f^{\prime \prime }>0$ 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 $e_{n+1}=f^{\prime \prime }({\xi }_n)e_n^2/[2f^{\prime }(x_n)]$, where $e_n=x_n-r$ and ${\xi }_n\in (r,x_n)$. Continuity proves the displayed positive asymptotic constant and quadratic convergence.