= Solution
The uniform stationarity equation and local stability condition are
$$
rM+uM^3+vM^5=h,\qquad V''(M)=r+3uM^2+5vM^4>0.
$$
For $u>0$, $h=0$, and small negative $r$, the stable nonzero solution satisfies $M^2=-r/u+O(r^2)$. Thus the <order parameter> vanishes continuously as $r\uparrow0$. The equilibrium <Landau free energy> has leading singular contribution $f_{\min}=-r^2/(4u)$ below the transition and zero above it. Its first thermal derivative is continuous, while its second derivative has a finite jump: the <mean-field approximation> predicts a <continuous phase transition> with no <latent heat>. The <magnetic susceptibility> follows by differentiating the stationarity equation:
$$
\chi=\frac{\partial M}{\partial h}=\frac{1}{r+3uM^2+5vM^4}.
$$
It diverges on approaching the ordinary <thermodynamic critical point>, and the quadratic <Landau scalar correlation length> behaves as $\xi\sim\sqrt{\kappa/|r|}$, with a different amplitude on the two sides.
For $u<0$, a nonzero stationary point at $h=0$ has $r+uM^2+vM^4=0$. Substitution into $V(M)-V(0)$ gives $-uM^4/4-vM^6/3$. Setting this to zero yields
$$
\boxed{M_{\mathrm{coex}}^2=-\frac{3u}{4v},\qquad r_{\mathrm{coex}}=\frac{3u^2}{16v}.}
$$
The finite jump of $M$ makes this a <first-order phase transition>. For a thermal path $r=r(T)$ with $r'(T)\ne0$ and the other coefficients fixed, $\partial f_{\min}/\partial T=r'(T)M^2/2$ jumps, giving an entropy discontinuity and nonzero <latent heat>. The <Landau free energy> itself remains continuous. The disordered minimum loses stability at $r=0$; the two nonzero stationary solutions merge at $r=u^2/(4v)$. These <spinodal points> bound <metastability>, but neither replaces the equilibrium <phase coexistence> condition.
Writing $r=a t$ with $a>0$ near an ordinary <thermodynamic critical point>, the zero-field equation gives $M\sim(-t)^{1/2}$. At $r=0$, small $h$ obeys $h=uM^3+O(M^5)$, hence $M\sim\operatorname{sgn}(h)|h|^{1/3}$. The ordinary <mean-field critical exponents> are therefore \b[$\beta=1/2$ and $\delta=3$]. These are <Landau theory> predictions, not dimension-independent values for an interacting theory below its <upper critical dimension>.
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