 Solution

For a homogeneous saddle-point approximation define $r=a(T-T_c)>0$. With the cubic source coupling, minimizing the Landau free energy givesIts curvature $2r+6b{m}^2$ is positive, so there is a unique local minimum and global minimum. The solution near $B=0$ hasbecause substituting this leading term makes the cubic self-interaction of order $B^9$. Therefore the cubic source coupling and third-order susceptibility giveThus the first nonzero response diverges like $(T-T_c{)}^{-1}$ from above. The factor $3!$ is essential: the coefficient of $B^3$ is not itself the third derivative. This is susceptibility to the specified cubic source variable, rather than the usual third-order response to a linearly coupled field.