Past exam of the mathematics course of the University of Cambridge / 2017 / ii / Paper 4 / 38B / i / Solution

For $0<V<\beta /(8{\ell }^2)$ there are two positive solutions $k_1\in (\ell ,\sqrt{3}\ell )$ and $k_2>\sqrt{3}\ell$, together with their negatives. The second derivative isEach negative stationary point has the conjugate contribution of its positive partner, because $A$ is real even and ${\Phi }_V$ is odd. ThereforeIf a leading amplitude vanishes, this gives a zero coefficient and a higher-order term supplies that branch's first nonzero contribution.