= Solution
At <phase coexistence>, the finite ordered <global minimum> has the same <free energy> as $m=0$. With $x=m^2>0$ and $f(0)=0$, stationarity and equal values give
$$
r+bx+cx^2=0,\qquad r+\frac b2x+\frac c3x^2=0.
$$
Subtracting eliminates $r$:
$$
\frac b2x+\frac{2c}{3}x^2=0,\qquad x_*=-\frac{3b}{4c}>0.
$$
Substitute back to get $r_*=3b^2/(16c)$, so
$$
\boxed{T_c^*=T_c+\frac{3b^2}{16ac},\qquad
m_{\rm jump}=\sqrt{-\frac{3b}{4c}}\quad(b<0).}
$$
The three degenerate equilibrium values at this temperature are $0$ and $\pm m_{\rm jump}$. A useful global check, avoiding confusion with a <spinodal point>, is the exact factorization
$$
\left.f(m)\right|_{r=r_*}=\frac c3m^2\left(m^2+\frac{3b}{4c}\right)^2\geq0.
$$
All three zeros are therefore <global minima>. Their curvatures are positive: $f''(0)=3b^2/(8c)$ and $f''(\pm m_{\rm jump})=3b^2/(2c)$.
The ordered local minimum first appears on cooling at $r=b^2/(4c)$, where the nonzero stationary roots merge. It becomes globally stable only at $r_*$; the disordered minimum remains <metastable> down to $r=0$. These different temperatures distinguish <phase coexistence> from loss of local stability.
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