= Solution
Let $t=(T-T_c)/T_c$, and suppose $r=r_t t$ with $r_t>0$ while $u>0$ and $\kappa>0$ remain finite at the transition. <Critical exponents> describe the leading singular powers: $M\sim(-t)^\beta$ below the transition, $\chi\sim|t|^{-\gamma}$, $\xi\sim|t|^{-\nu}$, $C_{V,s}\sim|t|^{-\alpha}$, $M(t=0)\sim\operatorname{sgn}(h)|h|^{1/\delta}$, and $G(R,t=0)\sim R^{-(D-2+\eta)}$.
The quartic <equation of state> $h=rM+uM^3$ immediately gives $M=\sqrt{-r/u}$ at zero field below the transition, so $\beta=1/2$. Differentiating with respect to the <conjugate field> gives $\chi^{-1}=r+3uM^2$, which is $r$ above and $2|r|$ below; therefore $\gamma=1$. At $r=0$, $M=(h/u)^{1/3}$ with its real sign, giving $\delta=3$. The quadratic <correlation function> has $\xi\propto|r|^{-1/2}$ and critical <Ornstein--Zernike correlation function> proportional to $q^{-2}$, giving $\nu=1/2$ and $\eta=0$.
Finally the minimized potential below the transition is $V_s=-r^2/(4u)$, while it is zero above. Its second <temperature> <derivative> is a finite step, so the mean-field <heat-capacity critical exponent> is $\alpha=0$. In particular,
$$
\boxed{(\alpha,\beta,\gamma,\delta,\nu,\eta)_{\rm MF}=(0,\tfrac12,1,3,\tfrac12,0).}
$$
These are derived <mean-field critical exponents>. In the fluctuation-dominated <critical region of a phase transition>, one instead derives exponents from the scaling <eigenvalues> of the <renormalization-group fixed point>, as in the next question. A zero heat-capacity power exponent can also accompany logarithms in other theories, so specifying the power alone does not completely describe that singularity.
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