= Solution
For $H=0$ and $a>0$, $F(M)=aM^2/2+bM^4/4$ is a symmetric single well. Its only minimum is
$$
\boxed{\langle M\rangle=0,\qquad F_{\min}=0.}
$$
The zero is relative to the smooth background omitted from the supplied expansion.
\Image[/past-exam-of-the-thermal-and-statistical-physics-course-of-the-university-of-cambridge/2013/paper-1-tsp2013-landau-minima.png]
{title=Original dimensionless quartic Landau free-energy sketches for the three printed cases, with their equilibrium minima marked}
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