= Solution
A <first-order phase transition> exchanges two distinct <global minima> of the <Landau free energy>. The equilibrium <free energy> remains continuous, but a first derivative such as the <magnetization> or <entropy> jumps. A temperature-driven transition generically has latent heat; crossing $h=0$ between symmetry-related ordered phases instead has a <magnetization> jump without requiring latent heat. A <continuous phase transition> has no such order-parameter jump and usually has a divergent <correlation length> and nonanalytic higher responses.
For $u>0$, $h=0$ and $r$ crossing zero, minimizing $V=rM^2/2+uM^4/4$ gives $M=0$ for $r>0$ and $M=\pm\sqrt{-r/u}$ for $r<0$. The minima merge continuously at zero. The curvature vanishes there, so the long-distance response becomes singular.
For $u<0<v$, put $z=M^2$. A nonzero stationary minimum satisfies $r+uz+vz^2=0$. Equality with the $M=0$ <free energy> gives
$$
\boxed{r=\frac{3u^2}{16v},\qquad M=0,\quad M=\pm\sqrt{-\frac{3u}{4v}}.}
$$
The finite separation demonstrates the <first-order phase transition>. <Metastability> occurs where a nonglobal <local minimum> persists; its disappearance at a <spinodal point> is not the equilibrium coexistence criterion.
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