The equilibrium magnetization is a <global minimum> of the <Landau free energy>. Its stationary values solve the <polynomial equation>
$$
\frac{\partial f}{\partial m}=4a_4m^3+6a_6m^5-B=0,
$$
and a local minimum must satisfy
$$
\frac{\partial^2f}{\partial m^2}=12a_4m^2+30a_6m^4\geq0.
$$
One compares the value of $f$ at every such local minimum and chooses the smallest. Since $a_6>0$, the <polynomial> tends to positive infinity as $|m|$ tends to infinity, so a global minimum exists.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 303 / 1 / a / i / Solution

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A polynomial equation has the form $p(x_1,\ldots,x_n)=0$ for a <polynomial> $p$. Its solutions are the points at which that polynomial vanishes.

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