= Solution
Write $a=a_0(T-T_c)$, $a_0>0$, at fixed $b\geq0$, and define the <critical exponent> by $|m|\sim(T_c-T)^\beta$ on the ordered side. For fixed $b>0$ and small negative $a$, the stationary equation gives $m^2=-a/b+O(a^2)$. Consequently
$$
\boxed{\beta=\frac12\quad\text{on the ordinary line }a=0,\ b>0.}
$$
At the <tricritical point> approached along the tuned path $b=0$, the stationary equation instead gives $m^4=-a/c$, so
$$
\boxed{\beta_{\mathrm{tri}}=\frac14\quad\text{at }a=b=0\text{ along }b=0.}
$$
For fixed $b<0$, the transition occurs before $a$ reaches zero and the <magnetization> jumps; it is not a second-order transition and has no vanishing-order-parameter exponent of this form. A path that does not keep the quartic coefficient tuned to zero generally crosses over to the ordinary result.
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