= Solution
Close to the critical temperature on the ordered side, both $m$ and $m/t$ are small. Expanding the <hyperbolic tangent> gives
$$
m=\frac mt-\frac{m^3}{3t^3}+O(m^5).
$$
For a nonzero solution, divide by $m$ and retain the cubic-order terms:
$$
1\simeq\frac1t-\frac{m^2}{3t^3},\qquad
\boxed{m\simeq t\sqrt{3(1-t)}\quad(0<1-t\ll1).}
$$
This <mean-field onset of two-state ferromagnetism> has <mean-field critical exponent> $1/2$: the curve reaches zero continuously with a square-root cusp below $t=1$, and remains exactly zero for all $t\geq1$. The factor $t$ is what the cubic truncation gives; higher orders also affect the coefficient at relative order $1-t$, so the robust leading asymptote is $m\sim\sqrt{3(1-t)}$.
\Image[/past-exam-of-the-quantum-condensed-matter-course-of-the-university-of-cambridge/2019-q6-molecular-field.png]
{title=Weiss self-consistency and the mean-field ferromagnetic onset}
{description=Original numerical solutions of $m=\tanh(m/t)$. The left panel shows a positive intersection below the critical temperature and none above it. The right panel plots the spontaneous magnitude and its near-critical cubic approximation; that approximation is shown only close to the transition.}
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