Asymptotics of a certain integral (Q756058)

The author proves that \[ \int^{\infty}_{0}\exp (tx-c(x))dx\sim \sqrt{\frac{2\pi}{c''(x_ 0)}}\exp (tx_ 0-c(x_ 0)),\quad t\to \infty, \] where \(x_ 0\) is defined by \(c'(x_ 0)=t\), and when the following conditions are satisfied: c is continuous on \([0,\infty [\); \(c''\) is positive and continuous on some \([a,\infty [\), \(a\geq 0\); \(\lim_{x\to \infty}c'(x)=\infty;\) for some positive function \(\alpha\) on \([b,\infty [\), \(b\geq 0\) with \(\alpha\) (x)\(\leq x\) we have that \(\lim_{x\to \infty}\alpha (x)\sqrt{c''(x)}=\infty,\) and for all \(\epsilon >0\) there is an \(A>0\) such that if \(| y-x| \leq \alpha (x)\) then \(| c''(y)-c''(x)| \leq \epsilon c''(x),\) whenever \(x\geq A\), \(y\geq 0\).