= Solution
Differentiating the <Landau theory> <free energy> gives
$$
F'=P[a(T-T_c)-bP+cP^2],\qquad F''=a(T-T_c)-2bP+3cP^2.
$$
Its stationary values are
$$
\boxed{P_0=0,\qquad P_\pm=\frac{b\pm\sqrt{b^2-4ac(T-T_c)}}{2c}}
$$
when the discriminant is nonnegative. At $P_0$, the curvature is $a(T-T_c)$: a local minimum above $T_c$ and a local maximum below it. Exactly at $T_c$, $F'''(0)=-2b\ne0$, so the origin is a stationary inflection point, not a stable minimum.
At a nonzero stationary point, eliminating $T-T_c$ yields $F''=P(2cP-b)$. Between $T_c$ and the upper <spinodal>, $P_+$ is a minimum and $P_-$ a maximum. Below $T_c$, $P_-<0$ becomes another local minimum, while $P_+>0$ is the lower minimum. Thus loss of local stability at $T_c$ is distinct from the equilibrium transition <temperature>.
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