= Solution
Discuss the zero-field phase boundaries, taking $h=0$ and writing $r=\mathcal A_2$, $q=\mathcal A_4$, $s=\mathcal A_6>0$. This qualification matters because a conjugate field generally rounds the zero-field continuous transition. Nonzero <stationary points> satisfy $r+qm^2+sm^4=0$.
For $q>0$, the <continuous phase transition> occurs at
$$
\boxed{r=0,\qquad q>0.}
$$
On the $r>0$ side the only stable phase is $m=0$; on the $r<0$ side the equilibrium <global minima> are $\pm m_0$ with $m_0^2\sim-r/q$. The <magnetization> therefore grows continuously from zero as $r$ changes sign.
For $q<0$, the <first-order phase transition> instead occurs while $r$ is positive. At <phase coexistence>, $m=0$ and $m=\pm m_0$ are equally deep <global minimum> points. With $a=m_0^2>0$, stationarity and equality of <free energies> give
$$
r+qa+sa^2=0,\qquad \frac12ra+\frac14qa^2+\frac16sa^3=0.
$$
Eliminating $r$ gives $-qa^2/4-sa^3/3=0$, hence
$$
\boxed{m_0^2=-\frac{3\mathcal A_4}{4\mathcal A_6},\qquad \mathcal A_2=\frac{3\mathcal A_4^2}{16\mathcal A_6},\qquad \mathcal A_4<0.}
$$
At this coexistence value, the <sextic even Landau potential> factors as $f(m)=s m^2(m^2+3q/(4s))^2/6\geq0$. This verifies that the three equal stationary values are <global minima>. Above this coexistence value of $r$ the disordered phase wins; below it the ordered phase wins, with a finite jump in the phase-selected <magnetization>. The values $r=0$ and $r=q^2/(4s)$ are the disordered and ordered <spinodal points>, respectively, not the coexistence boundary: the latter follows from the repeated root of $r+qa+sa^2=0$.
The two transition loci meet at the <tricritical point>,
$$
\boxed{\mathcal A_2=\mathcal A_4=0,\qquad \mathcal A_6>0.}
$$
Two independent control parameters such as $T$ and $g$ can tune these two conditions. In a generic local phase diagram their coefficient map has a <Jacobian matrix> of <rank> two at the intersection; unspecified functions $\mathcal A_2(T,g)$ and $\mathcal A_4(T,g)$ do not by themselves determine the geometric orientation of the boundaries. The sextic term stabilizes this intersection and the negative-quartic first-order side.
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