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 $f ^ { \prime \prime } \left( x _ { 0 } \right) \neq 0$. Under hypothesis $( \mathrm { H } )$, show that for all $\delta > 0$ such that $\delta < \min \left( x _ { 0 } - a , b - x _ { 0 } \right)$, we have the asymptotic equivalence, as $t \rightarrow + \infty$, $$\int _ { a } ^ { b } e ^ { t f ( x ) } \mathrm { d } x \sim \int _ { x _ { 0 } - \delta } ^ { x _ { 0 } + \delta } e ^ { t f ( x ) } \mathrm { d } x$$
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 $f ^ { \prime \prime } \left( x _ { 0 } \right) \neq 0$.
Under hypothesis $( \mathrm { H } )$, show that for all $\delta > 0$ such that $\delta < \min \left( x _ { 0 } - a , b - x _ { 0 } \right)$, we have the asymptotic equivalence, as $t \rightarrow + \infty$,
$$\int _ { a } ^ { b } e ^ { t f ( x ) } \mathrm { d } x \sim \int _ { x _ { 0 } - \delta } ^ { x _ { 0 } + \delta } e ^ { t f ( x ) } \mathrm { d } x$$