Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 331 / 1 / f / Solution

Put $x=k^2$ and $T=\lambda /\sigma$. Differentiating $Ra$ gives the exact stationarity equationWhen $T\gg 1$, the optimum has $x\gg {\pi }^2$, so $2x^3\sim {\pi }^2{T}^2$. Therefore