Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 60 / 3 / b / Solution

Put $x={\alpha }^2$ and $p={\pi }^2$. ThenThe minimizing condition isIts left side increases strictly for $x>0$ and the neutral curve diverges at both ends, so there is a unique global minimum. For large $Q$, $x\to \infty$ and the leading balance is $2x^3\sim Q{p}^2$. ThereforeThis is the strong-field wavenumber selection in magnetoconvection scaling. In a box of fixed width one compares the two admissible horizontal wavenumbers bracketing the continuum optimum.