Past exam of the mathematics course of the University of Cambridge / 2023 / ia / Paper 2 / 12F / d / Solution

Taylor's theorem about $1$ gives, for some $\xi \in [t,1]$,Since $G_X^{\prime \prime }$ is nondecreasing andthe stated inequality follows. At the fixed point $d<1$,Therefore the quadratic Galton-Watson extinction bound isThe denominator is at least $2(\mu -1)$ because $X$ is integer-valued and $\mathbb{E}[(X-1)(X-2)]\ge 0$, so $d^{\ast }\ge 0$.