= Solution
For positive $\lambda$, a fluctuation of <magnetization> produces a <Weiss molecular field> in the same direction, reinforcing alignment. The <mean-field self-consistency equation> is
$$
M=M_sB_J\!\left[\frac{g\mu_BJ\mu_0}{k_BT}(H_{\rm applied}+\lambda M)\right].
$$
Linearizing in the paramagnetic phase gives
$$
M=\frac CT(H_{\rm applied}+\lambda M),\qquad
\boxed{\chi=\frac C{T-\lambda C}=\frac C{T-T_C},\qquad T_C=\lambda C=\frac{\mu_0\lambda n(g\mu_B)^2J(J+1)}{3k_B}.}
$$
At zero applied field, the right-hand side's slope at $M=0$ is $T_C/T$. Below $T_C$ this exceeds one, and the saturation of the <Brillouin function> allows stable nonzero solutions of opposite sign: spontaneous <ferromagnetism>. Above $T_C$, only the zero-field solution $M=0$ remains stable. The stated linear susceptibility is the paramagnetic branch for $T>T_C$; it must not be continued as a negative susceptibility below the transition.
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