Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 4 / 35C / c / Solution

Minimization gives $m=0$ for $T>T_c$ and $m=\pm \sqrt{a(T_c-T)/b}$ below. ThusUsing $S=-F_T$ and $C=-T{F}_{TT}$, entropy is continuous but the low-temperature heat capacity has the additional $T{a}^2/(2b)$. The jump is $T_c{a}^2/(2b)$, so the transition is second order.