= Solution
The proposed <Landau free energy> has stationary equation $m[a(T-T_c)+bm^2]=0$. For $T>T_c$ its minimum is $m=0$. For $T<T_c$ the minima are $\boxed{m=\pm\sqrt{a(T_c-T)/b}}$, while zero becomes a maximum; at $T_c$ the positive quartic term keeps the free energy bounded below. Thus the two symmetry-related states, continuous transition and square-root onset agree with the <mean-field approximation>.
More explicitly the mean-field entropy expansion gives $F=F_0+N[k_B(T-T_c)m^2/2+k_BT m^4/12+O(m^6)]$ at zero field. Near $T_c$, replacing $T$ by $T_c$ in the quartic coefficient gives the positive constants $a=Nk_B$ and $b=Nk_BT_c/3$, and reproduces the preceding square-root amplitude. The polynomial is a local expansion near the transition, not a global expression at every temperature.
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