= Solution
The stationary points obey
$$
f'(m)=m(2a_2+3a_3m+4a_4m^2)=0.
$$
Nonzero stationary points exist when
$$
9a_3^2-32a_2a_4\geq0,
$$
so they first appear at the ordered-phase <spinodal point> $a_2=9a_3^2/(32a_4)$. The disordered state $m=0$ is locally stable for $a_2>0$ and loses that stability at $a_2=0$.
The actual phase boundary is found by requiring a nonzero stationary point $m_*$ to have the same free energy as $m=0$. Solving $f'(m_*)=0$ and $f(m_*)=0$ gives
$$
m_*=-\frac{a_3}{2a_4},
\qquad
a_2=\frac{a_3^2}{4a_4}.
$$
The disordered state is the global minimum for $a_2>a_3^2/(4a_4)$, the ordered state is the global minimum for $a_2<a_3^2/(4a_4)$, and they coexist at equality.
Since $a_3\ne0$, the order parameter jumps from zero to $-a_3/(2a_4)$ at coexistence. Hence this model has no <continuous phase transition> as the phases exchange stability, but it does have a <first-order phase transition> at the displayed positive value of $a_2$.
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