= Solution
<Critical exponents> describe leading singular powers as the reduced temperature $t$ or conjugate field approaches zero. With $\beta_m$ used for the <order-parameter critical exponent> to avoid confusion with inverse temperature, the definitions include
$$
C_s\sim |t|^{-\alpha},\quad M(t,0)\sim(-t)^{\beta_m},\quad\chi\sim|t|^{-\gamma},\quad\xi\sim|t|^{-\nu},\quad M(0,h)\sim\operatorname{sgn}(h)|h|^{1/\delta}.
$$
At criticality the <connected correlation function> behaves as $G(r)\sim r^{-(D-2+\eta)}$. These definitions refer to the singular part and allow additional logarithms at marginal dimensions.
For an ordinary scalar <Landau free energy>, the <equation of state> is $h=rM+uM^3$. Below the transition $M^2=-r/u$, giving $\beta_m=1/2$. Differentiating the <equation of state> gives $\chi^{-1}=r+3uM^2$: it equals $r$ above and $-2r$ below the transition, so $\gamma=1$. The quadratic fluctuation kernel is $\kappa k^2+r$ in the symmetric phase, giving <correlation length> $\xi=\sqrt{\kappa/r}$ and $\nu=1/2$. At zero mass the momentum-space <correlation function> is proportional to $k^{-2}$, so $\eta=0$. The minimized <free energy> and critical <equation of state> give $\alpha=0$ and $\delta=3$, as calculated in the root solution.
At the <tricritical point>, $u=0$ and $h=rM+vM^5$. Then $M\sim(-r)^{1/4}$, and $\chi^{-1}=r+5vM^4$ is $r$ above and $-4r$ below the transition. The gradient kernel still gives $\nu=1/2$ and $\eta=0$. Therefore
$$
\boxed{(\alpha,\beta_m,\gamma,\delta,\nu,\eta)_{\rm ordinary}=(0,1/2,1,3,1/2,0),\quad(\alpha,\beta_m,\gamma,\delta,\nu,\eta)_{\rm tricritical}=(1/2,1/4,1,5,1/2,0).}
$$
These are <mean-field critical exponents>. A systematic derivation incorporating fluctuations uses the <renormalization group>, extracts the relevant scaling eigenvalues, and differentiates the singular <free-energy density>, as in Question 3.
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