Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 2 / 36D / c / Solution

The function $f$ tends to $-\infty$ both as $r\to 0^{+}$ and as $r\to \infty$. Its only positive stationary point isand it is a maximum withTherefore $f$ has two positive simple roots exactly whenAt $r_c=1/(\sqrt{3}H)$,Since $f$ is positive precisely between its two positive roots, this proves