= Solution
A liquid-gas <thermodynamic critical point> is the endpoint of the coexistence curve where distinct liquid and gas densities merge. In the van der Waals illustration, write the original constants as $a_v,b_v$ to distinguish them from the later polynomial coefficients. The critical isotherm has a horizontal inflection in $p(V)$; setting its first two volume derivatives to zero gives
$$
\boxed{V_c=3Nb_v,\qquad k_BT_c=\frac{8a_v}{27b_v},\qquad p_c=\frac{a_v}{27b_v^2}.}
$$
Below this <temperature> the homogeneous van der Waals loop contains an unstable region; physical equilibrium instead separates into liquid and gas.<Classical Gases by David Tong>[Section 2.5.2: van der Waals equation and its critical point]{ref}
For the <critical point of a cubic density equation of state>, assume $b,c>0$. Its density derivatives are
$$
p_n=k_BT-bn+\tfrac c2n^2,\qquad p_{nn}=-b+cn.
$$
Because $n=N/V$ is a smooth monotone function of $V$ at finite positive density, a stationary inflection obeys $p_n=p_{nn}=0$ as well. Solving these equations and substituting back gives
$$
\boxed{n_c=\frac bc,\qquad k_BT_c=\frac{b^2}{2c},\qquad p_c=\frac{b^3}{6c^2}.}
$$
The symbols $b,c$ here have different units from the van der Waals excluded-volume parameter. With $x=n/n_c$, $t=T/T_c$, the isotherms become
$$
\frac p{p_c}=x^3-3x^2+3tx,\qquad\frac{d(p/p_c)}{dx}=3[(x-1)^2+t-1].
$$
The <spinodal densities of a cubic gas> delimit the two-extrema regime. The curves are monotone for $t>1$, have a stationary inflection at $(x,p/p_c)=(1,1)$ for $t=1$, and have two extrema $x=1\pm\sqrt{1-t}$ for $0<t<1$.
For the <critical-density compressibility of a cubic gas>, since $\kappa_T=(1/n)(\partial n/\partial p)_T$, evaluation at the critical density gives
$$
\boxed{\kappa_T(n_c,T)=\frac1{n_ck_B(T-T_c)}.}
$$
It diverges as $T\to T_c^+$ in this mean-field model. For $T<T_c$ it is negative on the homogeneous branch at $n_c$, violating <mechanical stability>: a <pressure> increase would expand rather than compress that branch.
\Image[/past-exam-of-the-physics-b-course-of-the-university-of-cambridge/2015/paper-2-critical-isotherms.png]
{title=Polynomial gas isotherms develop an unstable negative-slope region below criticality; the homogeneous compressibility becomes negative there}
{source=paper-2-critical-isotherms.py}
\b[The actual equilibrium below criticality is coexistence of two phases], with equal <pressure> and <chemical potential> and different densities. A sample's mean density can change by changing phase proportions. The unstable homogeneous branch is not a realizable equilibrium isotherm. The <Maxwell construction in density coordinates> requires care: the equal-area construction is performed in $p$ versus volume, not by using unweighted area in $p$ versus density. The truncated polynomial is an approximate mean-field equation, so its predictions need not give quantitative critical behaviour for a real gas.
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