Below is the graphical representation $\mathscr { C } _ { g }$ in an orthogonal coordinate system of a function $g$ defined and continuous on $\mathbb { R }$. The curve $\mathscr { C } _ { g }$ is symmetric with respect to the $y$-axis and lies in the half-plane $y > 0$.

For all $t \in \mathbb { R }$ we define:
$$G ( t ) = \int _ { 0 } ^ { t } g ( u ) \mathrm { d } u$$

\section*{Part A}
The justifications of the answers to the following questions may be based on graphical considerations.

\begin{enumerate}
  \item Is the function $G$ increasing on $[ 0 ; + \infty [$ ? Justify.
  \item Justify graphically the inequality $G ( 1 ) \leqslant 0.9$.
  \item Is the function $G$ positive on $\mathbb { R }$ ? Justify.
\end{enumerate}

In the rest of the problem, the function $g$ is defined on $\mathbb { R }$ by $g ( u ) = \mathrm { e } ^ { - u ^ { 2 } }$.

\section*{Part B}
\begin{enumerate}
  \item Study of $g$\\
a. Determine the limits of the function $g$ at the boundaries of its domain.\\
b. Calculate the derivative of $g$ and deduce the table of variations of $g$ on $\mathbb { R }$.\\
c. Specify the maximum of $g$ on $\mathbb { R }$. Deduce that $g ( 1 ) \leqslant 1$.
  \item We denote $E$ the set of points $M$ located between the curve $\mathscr { C } _ { g }$, the $x$-axis and the lines with equations $x = 0$ and $x = 1$. We call $I$ the area of this set.\\
We recall that:
$$I = G ( 1 ) = \int _ { 0 } ^ { 1 } g ( u ) \mathrm { d } u$$
We wish to estimate the area $I$ by the method called ``Monte-Carlo'' described below.
\begin{itemize}
  \item We choose a point $M ( x ; y )$ by randomly drawing its coordinates $x$ and $y$ independently according to the uniform distribution on the interval $[ 0 ; 1 ]$. It is admitted that the probability that the point $M$ belongs to the set $E$ is equal to $I$.
  \item We repeat $n$ times the experiment of choosing a point $M$ at random. We count the number $c$ of points belonging to the set $E$ among the $n$ points obtained.
  \item The frequency $f = \frac { c } { n }$ is an estimate of the value of $I$.\\
a. The figure below illustrates the method presented for $n = 100$. Determine the value of $f$ corresponding to this graph.\\
b. The execution of the algorithm below uses the Monte-Carlo method described previously to determine a value of the number $f$.\\
Copy and complete this algorithm.\\
$f , x$ and $y$ are real numbers, $n , c$ and $i$ are natural integers.\\
ALEA is a function that randomly generates a number between 0 and 1.
\begin{verbatim}
$c \leftarrow 0$
For $i$ varying from 1 to $n$ do:
        $x \leftarrow$ ALEA
        $y \leftarrow$ ALEA
    If $y \leqslant \ldots$ then
        $c \leftarrow \ldots$
    end If
end For
            $f \leftarrow \ldots$
\end{verbatim}
c. An execution of the algorithm for $n = 1000$ gives $f = 0.757$.\\
Deduce a confidence interval, at the 95\% confidence level, for the exact value of $I$.
\end{itemize}
\end{enumerate}

\section*{Part C}
We recall that the function $g$ is defined on $\mathbb { R }$ by $g ( u ) = \mathrm { e } ^ { - u ^ { 2 } }$ and that the function $G$ is defined on $\mathbb { R }$ by:
$$G ( t ) = \int _ { 0 } ^ { t } g ( u ) \mathrm { d } u$$
We propose to determine an upper bound for $G ( t )$ for $t \geqslant 1$.

\begin{enumerate}
  \item A preliminary result.\\
It is admitted that, for all real $u \geqslant 1$, we have $g ( u ) \leqslant \frac { 1 } { u ^ { 2 } }$.\\
Deduce that, for all real $t \geqslant 1$, we have:
$$\int _ { 1 } ^ { t } g ( u ) \mathrm { d } u \leqslant 1 - \frac { 1 } { t }$$
  \item Show that, for all real $t \geqslant 1$,
$$G ( t ) \leqslant 2 - \frac { 1 } { t }$$
What can we say about the possible limit of $G ( t )$ as $t$ tends to $+ \infty$ ?
\end{enumerate}