= Solution
In a <first-order phase transition>, two distinct <global minima> of the <Landau free energy> have equal values and exchange stability. The equilibrium <order parameter> jumps, and a first derivative of the equilibrium <free energy> is discontinuous. Depending on the direction through the <phase diagram>, that derivative can be <magnetization>, <entropy> or another conjugate quantity; a first-order transition need not have latent heat along every chosen path.
In a <continuous phase transition>, the equilibrium <order parameter> changes continuously while the curvature at the critical minimum vanishes. For $u>0$ and $h=0$, $M=0$ is stable for $r>0$, but for $r<0$ the stable minima have $M\simeq\pm\sqrt{-r/u}$. The <order parameter> therefore goes continuously to zero at $r=0$; the inverse <magnetic susceptibility> vanishes there.
For $u<0$, the sextic term stabilizes the <Landau free energy>. The nonzero minima can become favorable while $M=0$ still has positive curvature. At $r=3u^2/(16v)>0$ they coexist with $M=0$, so the <order parameter> jumps to magnitude $\sqrt{-3u/(4v)}$. Finding a stationary point alone does not locate the transition: \b[equilibrium requires comparing the values of all competing minima]. Spinodal boundaries describe loss of local stability, not coexistence.
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