= Solution
Keep the fixed <stellar homology> profile, so that $\rho_c=C_\rho M/R^3$ and $P_c=C_PGM^2/R^4$. Solving the partially degenerate <equation of state> for the central <temperature> gives
$$
T_c=\frac{\mu_i}{\mathcal R}
\left(\frac{P_c}{\rho_c}-K_{\rm nr}\rho_c^{2/3}\right)
=\frac{\mu_i}{\mathcal R}
\left(A\frac{GM}{R}-B K_{\rm nr}\frac{M^{2/3}}{R^2}\right),
$$
where $A=C_P/C_\rho$ and $B=C_\rho^{2/3}$ are positive constants. The <ion mean molecular weight> is distinct from the total-particle <mean molecular weight> used in part (a).
Set $y=1/R$. The expression in parentheses is a concave quadratic in $y$. Its maximum occurs at
$$
y_{\max}=\frac{AG}{2BK_{\rm nr}}M^{1/3},\qquad
R_{\max}=\frac{2BK_{\rm nr}}{AG}M^{-1/3}.
$$
Substitution gives the \b[contraction-limited central temperature]
$$
\boxed{T_{\max}=\frac{A^2G^2}{4BK_{\rm nr}\mathcal R}\,\mu_iM^{4/3}
\ \propto\ \mu_iM^{4/3}.}
$$
At this point <electron degeneracy pressure> supplies half the required central pressure. Further contraction increasingly goes into the density-dependent degeneracy term rather than heating the ions. The formula applies along the positive-temperature part of the idealized homologous sequence; a negative value beyond it is not a physical temperature.
For sufficiently small mass, this maximum never reaches the temperature needed for sustained <hydrogen burning>. Such an object stays below the <hydrogen-burning minimum mass> and becomes a cooling <brown dwarf>, rather than an ordinary <main sequence> star. Some such objects burn their initial <deuterium>, but that finite reservoir does not provide lasting <hydrogen burning>.
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