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 (H), show the asymptotic equivalence, as $t \rightarrow + \infty$, $$\int _ { a } ^ { b } e ^ { t f ( x ) } \mathrm { d } x \sim e ^ { t f \left( x _ { 0 } \right) } \sqrt { \frac { 2 \pi } { t \left| f ^ { \prime \prime } \left( x _ { 0 } \right) \right| } }$$
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 (H), show the asymptotic equivalence, as $t \rightarrow + \infty$,
$$\int _ { a } ^ { b } e ^ { t f ( x ) } \mathrm { d } x \sim e ^ { t f \left( x _ { 0 } \right) } \sqrt { \frac { 2 \pi } { t \left| f ^ { \prime \prime } \left( x _ { 0 } \right) \right| } }$$