= Solution
A <first-order phase transition> has a discontinuity in a first derivative of the equilibrium <free energy>, such as <entropy> or the <order parameter>; a thermal first-order transition normally has <latent heat>. At a <continuous phase transition> the <order parameter> goes continuously to zero and there is no <latent heat>, although response functions and higher derivatives of the <free energy> can be singular.
For the symmetric scalar <Landau free energy> at zero conjugate field, minimize
$$
V(M)=\frac r2M^2+\frac u4M^4+\frac v6M^6,\qquad
V'(M)=M(r+uM^2+vM^4).
$$
If $u>0$, the sixth-order term is unnecessary near $r=0$. For $r>0$ the minimum is $M=0$, whereas for $r<0$ there are two symmetry-related minima with $M^2=-r/u$ to leading order. The <order parameter> vanishes continuously as $r\uparrow0$. The equilibrium <free-energy density> changes from zero to $-r^2/(4u)$, so its first temperature derivative is continuous and its second derivative has a jump. This is the ordinary mean-field <continuous phase transition>.
If $u<0$, keep $v>0$ for stability. A nonzero stationary <order parameter>, with $s=M^2$, obeys $r+us+vs^2=0$. For <phase coexistence> with $M=0$ its <free energy> must also vanish. Substituting $r=-us-vs^2$ into $V$ gives
$$
V=-\frac u4s^2-\frac v3s^3.
$$
The nonzero solution of $V=0$ is therefore
$$
\boxed{M_t^2=-\frac{3u}{4v},\qquad r_t=\frac{3u^2}{16v}.}
$$
The competing minima exchange stability at positive $r_t$, with a finite jump of the <order parameter>. This is a <first-order phase transition>. The <metastability> region is also visible: nonzero stationary points first appear at the <spinodal> $r=u^2/(4v)$, and the disordered local minimum loses stability at $r=0$.
For example, holding $u,v$ fixed and taking $r=a(T-T_0)$, the ordered phase has <entropy> lower than the disordered phase by $aM_t^2/2$ at coexistence. Heating through the transition absorbs <latent heat> per volume $T_taM_t^2/2$. More generally the temperature dependence of all coefficients contributes to this <entropy> difference. With no inversion symmetry, a cubic term can also produce a <first-order phase transition>, as comparison of the competing minima of the <Landau free energy> shows.
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