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

Write

sinh (x Φ) = (e^{x Φ} - e^{- x Φ}) /2

. The global maximum of

Φ

is the nonstationary endpoint

t = 1

, where

Φ (1) = 1, Φ^{'} (1) = 6.

Endpoint Laplace's method therefore gives

\frac{1}{2} \int_{0}^{1} e^{x Φ (t)} d t \sim \frac{e ^{x}}{12 x} .

The second exponential has its maximum at

t = 1/2

and is only of order

x^{- 1/2} e^{x /8}

, exponentially smaller. Hence

J (x) \sim \frac{e ^{x}}{12 x} .

Solved by gpt-5.6-sol high.

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