= Solution
Take $a=a_0(T-T_c)$ and hold $b,c$ fixed. The <heat capacity> comes from the minimized <free energy>, $C=-T\,d^2F_{\min}/dT^2$, with any smooth background added separately. For $b>0$ and $T<T_c$,
$$
F_{\min}-F_{\mathrm{reg}}=-\frac{a^2}{4b}+O(a^3),\qquad
C_{\mathrm{sing},-}=\frac{T a_0^2}{2b}+O(T_c-T).
$$
Above $T_c$, $m=0$ and the polynomial contributes zero. There is a finite jump, rather than a divergence: the conventional ordinary <mean-field critical exponents> are
$$
\boxed{\alpha_-=\alpha_+=0\quad(b>0),\qquad \Delta C=\frac{T_ca_0^2}{2b}.}
$$
The upper branch has zero order-parameter contribution; a finite regular <heat capacity> has exponent zero.
Along the tuned tricritical path $b=0$, the ordered minimum gives
$$
F_{\min}-F_{\mathrm{reg}}=-\frac{(-a)^{3/2}}{3\sqrt c},\qquad
\boxed{C_{\mathrm{sing},-}=\frac{T a_0^2}{4\sqrt c\sqrt{-a}}
=\frac{T a_0^{3/2}}{4\sqrt c}(T_c-T)^{-1/2},\quad \alpha_-^{\mathrm{tri}}=\frac12.}
$$
\b[Upper-side qualification:] the <vanishing disordered-side heat-capacity amplitude in Landau theory> matters here: for $b=0$ and $a>0$, the minimizing solution is still exactly $m=0$, so this saddle-point polynomial gives $F_{\min}-F_{\mathrm{reg}}=0$ and $C_{\mathrm{sing},+}=0$. A zero function cannot determine a nonzero-amplitude power-law exponent. For example, adding the allowed smooth background $F_{\mathrm{reg}}=-C_0T\log(T/T_*)$, with $C_0>0$, leaves the entire <phase diagram> unchanged but makes the upper-side <heat capacity> exactly $C_0$, not a $(T-T_c)^{-1/2}$ divergence.
One convention assigns the tricritical scaling exponent $\alpha=1/2$ to both sides, writing $C_{\mathrm{sing},\pm}=A_\pm|T-T_c|^{-1/2}$ with $A_+=0$ at this <mean-field approximation> saddle. If the source's literal $C\sim|T-T_c|^{-\alpha_+}$ is meant to have a nonzero leading coefficient, the given polynomial establishes \b[no upper-side tricritical singular exponent]; with a nonzero smooth background its leading upper-side exponent is zero. A nonzero upper-side singularity requires fluctuation information beyond this minimization. This distinction prevents claiming an upper-side divergence that the displayed <Landau free energy> does not produce.
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