Past exam of the mathematics course of the University of Cambridge / 2024 / ib / Paper 4 / 18H / b / Solution

If $x_k\ge 1$, then positivity and the arithmetic-geometric mean inequality giveThus all iterates remain in $[1,\infty )$.

Consider the strictly convex functionon $[1,\infty )$. Since $f^{\prime }(x)=x^2-a$ and $f^{\prime \prime }(x)=2x$, its Newton minimization step isIts unique minimizer is $x^{\ast }=\sqrt{a}$.