= Solution
Write the optimized kernel as
$$
J(q)=\bar a+\kappa q^2+\gamma q^4,
\qquad
\bar a=a+\Delta.
$$
Near the soft-mode sphere, the supplied asymptotic result says
$$
I(\bar a)\sim C(\bar a-a_c)^{-1/2},
\qquad
\Delta\sim
6g\sqrt{\frac{2C}{\pi}}\,
(\bar a-a_c)^{-1/4}.
$$
Thus the <self-consistency equation>
$$
\bar a-a_c
=a-a_c+\Delta
$$
cannot reach $\bar a=a_c$ for any $g>0$: the fluctuation correction diverges first. The isotropic <static structure factor> therefore remains finite, and the isotropic state never undergoes the continuous Gaussian instability predicted in part (a).
The modulated minimum from part (b), however, has negative free energy for sufficiently low $a$ and amplitude $A_*=\pi(a_c-a)/(8g)$. For small positive $g$, its free energy must cross that of the still locally stable isotropic variational state. At the crossing the isotropic inverse susceptibility is positive and the smectic amplitude is nonzero. The order parameter consequently jumps, which is the <Brazovskii fluctuation-induced first-order transition>.
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