= Solution
Use the stable parameters $b,\kappa>0$. The bulk stationary points satisfy $f'(\phi)=a\phi+b\phi^3=0$. For $a<0$, $\phi=0$ has negative curvature, while the two minima are
$$
\boxed{\phi=\pm\phi_B,\qquad\phi_B=\sqrt{-a/b}.}
$$
Both have $f(\pm\phi_B)=-a^2/(4b)$ and $f''(\pm\phi_B)=-2a>0$. To establish minimization at fixed total composition, write
$$
f(\phi)=-\frac{a^2}{4b}+\frac b4(\phi^2-\phi_B^2)^2,
\qquad F\geq-\frac{Va^2}{4b}.
$$
For mean composition $\bar\phi$ with $|\bar\phi|<\phi_B$, a mixture of the two minima with volume fraction $\lambda=(\bar\phi+\phi_B)/(2\phi_B)$ in the positive phase attains this lower bound in the bulk. Macroscopic slabs have only a subextensive interfacial excess <free energy>, so the bound is attained per unit volume in the <thermodynamic limit>. Thus constrained minimization gives <phase coexistence> in the <mean-field approximation>; at the endpoints only one phase is needed.
The two equilibrium conditions also follow directly by <constrained optimization>. Introduce a <Lagrange multiplier> $\mu$ for $\lambda\phi_++(1-\lambda)\phi_-=\bar\phi$ in the mixture's bulk <free-energy density>. Varying $\phi_+$ and $\phi_-$ gives $f'(\phi_+)=f'(\phi_-)=\mu$; varying $\lambda$ gives $f(\phi_+)-\mu\phi_+=f(\phi_-)-\mu\phi_-$. These equate the <chemical potentials> and <pressures>, respectively.
Equivalently, their bulk <chemical potentials> are both zero and their <pressures> are equal:
$$
\boxed{\mu=f'(\pm\phi_B)=0,\qquad\Pi=\mu\phi-f(\phi)=\frac{a^2}{4b}.}
$$
These are precisely the conditions of the <common-tangent construction for phase coexistence>: the horizontal supporting tangent touches both minima. Equal <chemical potentials> prevent net composition exchange, and equal <pressure> gives mechanical equilibrium at a flat interface.
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