For the function $f ( x ) = e ^ { - x } \int _ { 0 } ^ { x } \sin \left( t ^ { 2 } \right) d t$, which of the following statements are correct? [4 points]
ㄱ. $f ( \sqrt { \pi } ) > 0$ ㄴ. There exists at least one $a$ in the open interval $( 0 , \sqrt { \pi } )$ such that $f ^ { \prime } ( a ) > 0$. ㄷ. There exists at least one $b$ in the open interval $( 0 , \sqrt { \pi } )$ such that $f ^ { \prime } ( b ) = 0$.
(1) ㄱ
(2) ㄷ
(3) ㄱ, ㄴ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ

If $a$ and $b$ are two real numbers, we denote $K _ { a , b }$ the function defined for all real $t$ by $K _ { a , b } ( t ) = \begin{cases} \frac { \mathrm { e } ^ { \mathrm { i } t b } - \mathrm { e } ^ { \mathrm { i } t a } } { 2 \mathrm { i } t } & \text { if } t \neq 0 , \\ \frac { b - a } { 2 } & \text { if } t = 0 . \end{cases}$ Let $N$ be a natural integer and let $F _ { N }$ be the function defined, for all real $x$, by $F _ { N } ( x ) = \int _ { - N } ^ { N } K _ { a , x } ( t ) \mathrm { d } t$. Show that $F _ { N }$ is of class $C ^ { 1 }$ on $\mathbb { R }$ and that, for all real $x , F _ { N } ^ { \prime } ( x ) = N \operatorname { sinc } ( N x )$.

Justify that the function $\psi_n$ is differentiable on $\mathbb{R}_+$ and that $\psi_n' = m_n$.
We admit that, when $\psi$ is differentiable on $\mathbb{R}_+^*$, then $(\lim \psi_n)' = \lim \psi_n'$, that is $\psi' = m$, on $\mathbb{R}_+^*$.