= Solution
A <first-order phase transition> exchanges distinct <global minima> of the <Landau free energy>; the <order parameter> jumps and a thermal crossing generically has <latent heat>. A <continuous phase transition> has minima merging continuously and a diverging <correlation length>. A local loss of stability at a <spinodal point> is not the equilibrium coexistence condition.
For the <sextic even Landau potential>, write $y=M^2$. At a nonzero zero-field <stationary point>, $r+uy+vy^2=0$. To coexist with $M=0$, also require $ry/2+uy^2/4+vy^3/6=0$. Eliminate $r$ to obtain $y=-3u/(4v)$ and $r=3u^2/(16v)$, which requires $u<0$. On this locus
$$
V(M)=\frac v6M^2\left(M^2+\frac{3u}{4v}\right)^2\geq0,
$$
proving that all three equal minima are globally stable. Their curvatures are $3u^2/(16v)$ at zero and $3u^2/(4v)$ at the nonzero minima, so the <correlation length> is finite when $u<0$ is fixed. For $u>0$, by contrast, the nonzero minimum approaches zero as $r\uparrow0$, giving the ordinary continuous transition. At fixed $u>0$ and $r<0$, crossing $h=0$ exchanges the two ordered minima discontinuously even though the zero-field thermal transition at $r=0$ is continuous.
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