Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 2 / 32A / c / Solution

Write $\sinh (x\Phi )=(e^{x\Phi }-e^{-x\Phi })/2$. The global maximum of $\Phi$ is the nonstationary endpoint $t=1$, whereEndpoint Laplace's method therefore givesThe second exponential has its maximum at $t=1/2$ and is only of order $x^{-1/2}{e}^{x/8}$, exponentially smaller. Hence