= Solution
The coarse-grained <order parameter> is a <real scalar field> because the microscopic spin has one component. At $B=0$, <spin inversion symmetry> takes $m\mapsto-m$, so an analytic local <Landau free energy> contains only even powers. Near a <continuous phase transition>, the quadratic coefficient changes sign linearly, represented by $a(T-T_c)$ with $a>0$. A positive quartic coefficient stabilizes the density when the quadratic term becomes negative, giving the two <minima> $m=\pm\sqrt{a(T_c-T)/b}$ below $T_c$.
The ferromagnetic interaction favours neighbouring spins pointing alike, so spatial variation costs energy; its leading long-wavelength form is the positive <gradient> term $c(\partial_xm)^2$. The source term $-Bm$ favours alignment with the applied <magnetic field> and explicitly breaks <spin inversion symmetry> at fixed nonzero $B$. Finally $f_0(T)$ is the spin-independent background <free-energy density>.
The quartic sign can also be checked from part (b): its expansion contains
$$
\frac{F_{\rm MF}}N=-\frac{\log3}{\beta}+\left(2J-\frac{16\beta J^2}{3}\right)S^2+\frac{64\beta^3J^4}{9}S^4+O(S^6),
$$
whose quartic coefficient is positive. This is consistent with the proposed <Landau-Ginzburg theory>, not a tricritical truncation requiring a stabilizing sixth-order term. The microscopic <square lattice> is two-dimensional; the written gradient describes a profile depending on one coordinate. For general profiles, replace it by $c[(\partial_xm)^2+(\partial_ym)^2]$.
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