= Solution
Fix $c>0$. <Stationary points> satisfy $m(a+bm^2+cm^4)=0$. With $x=m^2$, the nonzero candidates are
$$
x_\pm=\frac{-b\pm\sqrt{b^2-4ac}}{2c}.
$$
Keep only real positive roots. At such a point,
$$
F''(m)=2x(b+2cx),
$$
so $x_+$ gives <local minima> and $x_-$ gives local maxima when both exist. The solution $m=0$ is a <local minimum> for $a>0$. To choose the phase, compare <free energies>: using the <stationary point> relation $a=-bx-cx^2$ gives
$$
F(m)-F(0)=-\frac b4x^2-\frac c3x^3.
$$
For $b>0$, this expression is negative whenever a nonzero stationary minimum exists. The <paramagnetism> phase is stable for $a>0$, and the <ferromagnetism> phase with $m=\pm\sqrt{x_+}$ is stable for $a<0$. The <phase transition> at $a=0$, $b>0$, is continuous and second order.
For $b<0$, equality with $F(0)$ requires $x=-3b/(4c)$. Substitution into the stationary equation gives the first-order coexistence curve
$$
\boxed{a_{\mathrm t}=\frac{3b^2}{16c},\quad b<0,\qquad
m_{\mathrm t}^2=-\frac{3b}{4c}.}
$$
The disordered phase is the <global minimum> above this curve, and the ordered phase is the <global minimum> below it. At coexistence the factorization
$$
\boxed{F(m)=\frac c6m^2\left(m^2+\frac{3b}{4c}\right)^2}
$$
exhibits three equally low minima, $m=0$ and $m=\pm\sqrt{-3b/(4c)}$, proving the discontinuous jump.
The ordered <spinodal> is $a=b^2/(4c)$, $b<0$, where its <local minimum> meets the intervening local maximum. The disordered <spinodal> is $a=0$, $b<0$. Thus the ordered phase is metastable for $a_{\mathrm t}<a<b^2/(4c)$; the disordered phase is metastable for $0<a<a_{\mathrm t}$. These <spinodals> are limits of <metastability>, not equilibrium transition curves.
The first-order curve meets the continuous transition line at the <tricritical point> $(a,b)=(0,0)$. On the specially tuned path $b=0$, the equilibrium ordered solution is $m=\pm(-a/c)^{1/4}$ for $a<0$ and tends continuously to zero at the origin.
\Image[/past-exam-of-the-theoretical-physics-major-option-course-of-the-university-of-oxford/2018-sextic-landau-phase-diagram.png]
{title=<Phase diagram> of the sextic <Landau free energy>, including coexistence, <spinodals> and the <tricritical point>}
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