= Solution
A <tricritical point> occurs when a first-order boundary meets a continuous-transition boundary. In the spin-inversion-symmetric sextic <Landau free energy>, two independent parameters must tune $r=u=0$, while $v>0$ keeps the potential stable. In the $h=0$ plane, the continuous boundary is $r=0,u>0$, and the first-order boundary is $r=3u^2/(16v),u<0$. They meet at the tricritical point; the order-parameter jump on the first-order side tends to zero as $u\uparrow0$. The first panel shows this <phase diagram>, including the metastability limits derived above.
Along the tuned line $u=0,h=0$, the equation of state is $M(r+vM^4)=0$. Thus $M=(-r/v)^{1/4}$ on a selected ordered branch, rather than the ordinary square-root behavior. Differentiation gives inverse <magnetic susceptibility> $r$ above and $4|r|$ below, while at criticality $h=vM^5$. Hence the <mean-field tricritical exponent calculation> yields $\beta=1/4$, $\gamma=1$, $\delta=5$; the minimized potential $-|r|^{3/2}/(3\sqrt v)$ also gives $\alpha=1/2$. These tricritical powers require tuning the quartic coefficient, not merely varying a generic temperature path. The ordinary scalar exponents requested separately are calculated next.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2004/iii/paper-46-phases.png]
{title=Scalar sextic Landau phase diagram with a tricritical point, and quartic metastable hysteresis branches}
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