Solution
= Solution
Differentiating the mean-field free energy and imposing stationarity gives
$$
0=Jqm-Jq\frac{\sinh(\beta Jqm)}{1+\cosh(\beta Jqm)}.
$$
The <hyperbolic-function identity> $\sinh x/(1+\cosh x)=\tanh(x/2)$ turns this into the <self-consistency equation>
$$
\boxed{m=\tanh\!\left(\frac{Jqm}{2T}\right)}.
$$