= Solution
On a tricritical thermal path, the quartic scaling variable $X$ tends to zero, and at zero field $Y=0$. On the ordered side the <tricritical crossover scaling> function tends to $\Phi_-(0,0)=-1/3$. Consequently
$$
f_{\rm s}\sim-\frac{|\mathcal A_2|^{3/2}}{3\mathcal A_6^{1/2}}\sim-(T_c-T)^{3/2},
$$
provided the quadratic coefficient has a nonzero thermal slope and the sextic coefficient has a finite positive limit. Since the <heat-capacity critical exponent> satisfies $f_{\rm s}\sim|T-T_c|^{2-\alpha}$,
$$
\boxed{2-\alpha=\frac32,\qquad \alpha=\frac12.}
$$
Equivalently, twice differentiating this ordered-side <free-energy density> produces the $(T_c-T)^{-1/2}$ singularity. The scaling path and branch qualifications are the same as for the direct <Landau theory> calculation: a nonzero fixed positive quartic coefficient would instead send $X\to\infty$ and yield the ordinary transition of part (e).
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