Let $a < b$ be two real numbers and $f : [a,b] \rightarrow \mathbb{R}$ be an infinitely differentiable function. Let us call (H) the following hypothesis: there exists a unique point $x_0 \in [a,b]$ where $f$ attains its maximum, we have $a < x_0 < b$, and $f''(x_0) \neq 0$. Under hypothesis (H), show that for all $\delta > 0$ such that $\delta < \min(x_0 - a, b - x_0)$, we have the asymptotic equivalence, as $t \rightarrow +\infty$, $$\int_a^b e^{tf(x)} \mathrm{d}x \sim \int_{x_0 - \delta}^{x_0 + \delta} e^{tf(x)} \mathrm{d}x.$$
Let $a < b$ be two real numbers and $f : [a,b] \rightarrow \mathbb{R}$ be an infinitely differentiable function. Let us call (H) the following hypothesis: there exists a unique point $x_0 \in [a,b]$ where $f$ attains its maximum, we have $a < x_0 < b$, and $f''(x_0) \neq 0$.
Under hypothesis (H), show that for all $\delta > 0$ such that $\delta < \min(x_0 - a, b - x_0)$, we have the asymptotic equivalence, as $t \rightarrow +\infty$,
$$\int_a^b e^{tf(x)} \mathrm{d}x \sim \int_{x_0 - \delta}^{x_0 + \delta} e^{tf(x)} \mathrm{d}x.$$