= Solution
A <first-order phase transition> has a discontinuity in a first derivative of the equilibrium <free energy>, such as the <entropy> or <order parameter>. It can have latent heat when the <entropy> jumps. A <continuous phase transition> has a continuously vanishing <order parameter> and no latent heat, with singular higher derivatives and a diverging <correlation length>. The <LG theory> compares <global minima>, not merely the points where a local minimum loses stability.
For the uniform quartic <Landau free energy>
$$
V(m)=\tfrac12rm^2+\tfrac14um^4-hm,\qquad u>0,
$$
the <equation of state> is $rm+um^3=h$. At zero field the stable minimum is $m=0$ for $r>0$, and $m=\pm\sqrt{-r/u}$ for $r<0$. Thus tuning $r$ through zero gives a \b[continuous transition]. For a fixed $r<0$, varying $h$ through zero instead switches between the two ordered minima and makes $m$ jump, giving a <first-order phase transition> in the <conjugate field>.
A temperature-like first-order transition can occur at zero field when $u<0$ and a positive sextic term $vm^6/6$ stabilizes the potential. Write $q=m^2$. Stationarity of a nonzero phase gives $r+uq+vq^2=0$, while equality with $V(0)=0$ gives $rq/2+uq^2/4+vq^3/6=0$. Solving these two conditions yields
$$
\boxed{q_{\rm coex}=-\frac{3u}{4v},\qquad r_{\rm coex}=\frac{3u^2}{16v}\quad(u<0).}
$$
The <order parameter> jumps from zero to $\pm\sqrt{q_{\rm coex}}$. At this point
$$
V(m)=\frac v6m^2\left(m^2+\frac{3u}{4v}\right)^2\geq0,
$$
so the competing stationary points are genuinely <global minima>. This <phase coexistence> condition differs from the <spinodal points> $r=0$ and $r=u^2/(4v)$, which mark loss or creation of local stability, not equilibrium coexistence.
If the symmetry permits a cubic term $wm^3/3$, a positive quartic coefficient does not preclude a first-order transition. For $V=rm^2/2+wm^3/3+um^4/4$ with $w\ne0,u>0$, stationarity and coexistence give $m=-2w/(3u)$ and $r=2w^2/(9u)$. This is the <first-order transition in a cubic-quartic Landau potential>; the symmetry restriction on the expansion is therefore part of the prediction.
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