Past exam of the mathematics course of the University of Cambridge / 2025 / ia / Paper 2 / 2C / b

The fixed points are $x=-2\pm 2\sqrt{1+b}$, so they exist for $b\ge -1$. A fixed point of an iteration is stable when $|g^{\prime }(x)|<1$; here this is $|x|<2$. Only the plus root can satisfy it, and it does precisely for