\section*{Exercise 1 -- Common to all candidates}

\section*{Part A}
We consider the function $h$ defined on the interval $[ 0 ; + \infty [$ by:
$$h ( x ) = x \mathrm { e } ^ { - x }$$

\begin{enumerate}
  \item Determine the limit of the function $h$ at $+ \infty$.
  \item Study the variations of the function $h$ on the interval $[ 0 ; + \infty [$ and draw up its table of variations.
  \item The objective of this question is to determine a primitive of the function $h$.\\
a. Verify that for every real number $x$ belonging to the interval $[ 0 ; + \infty [$, we have:
$$h ( x ) = \mathrm { e } ^ { - x } - h ^ { \prime } ( x )$$
where $h ^ { \prime }$ denotes the derivative function of $h$.\\
b. Determine a primitive on the interval $[ 0 ; + \infty [$ of the function $x \longmapsto \mathrm { e } ^ { - x }$.\\
c. Deduce from the two previous questions a primitive of the function $h$ on the interval $[ 0 ; + \infty [$.
\end{enumerate}

\section*{Part B}
We define the functions $f$ and $g$ on the interval $[ 0 ; + \infty [$ by:
$$f ( x ) = x \mathrm { e } ^ { - x } + \ln ( x + 1 ) \quad \text { and } \quad g ( x ) = \ln ( x + 1 )$$
We denote $\mathcal { C } _ { f }$ and $\mathcal { C } _ { g }$ the respective graphical representations of the functions $f$ and $g$ in an orthonormal coordinate system.

\begin{enumerate}
  \item For a real number $x$ belonging to the interval $[ 0 ; + \infty [$, we call $M$ the point with coordinates $( x ; f ( x ) )$ and $N$ the point with coordinates $( x ; g ( x ) )$: $M$ and $N$ are therefore the points with abscissa $x$ belonging respectively to the curves $\mathcal { C } _ { f }$ and $\mathcal { C } _ { g }$.\\
a. Determine the value of $x$ for which the distance $MN$ is maximum and give this maximum distance.\\
b. Place on the graph provided in the appendix the points $M$ and $N$ corresponding to the maximum value of $MN$.
  \item Let $\lambda$ be a real number belonging to the interval $[ 0 ; + \infty [$. We denote $D _ { \lambda }$ the region of the plane bounded by the curves $\mathcal { C } _ { f }$ and $\mathcal { C } _ { g }$ and by the lines with equations $x = 0$ and $x = \lambda$.\\
a. Shade the region $D _ { \lambda }$ corresponding to the value $\lambda$ proposed on the graph in the appendix.\\
b. We denote $A _ { \lambda }$ the area of the region $D _ { \lambda }$, expressed in square units. Prove that:
$$A _ { \lambda } = 1 - \frac { \lambda + 1 } { \mathrm { e } ^ { \lambda } } .$$
c. Calculate the limit of $A _ { \lambda }$ as $\lambda$ tends to $+ \infty$ and interpret the result.
  \item We consider the following algorithm:
\begin{verbatim}
Variables:
    $\lambda$ is a positive real number
    $S$ is a real number strictly between 0 and 1.
Initialization:
    Input $S$
    $\lambda$ takes the value 0
Processing:
    While $1 - \frac { \lambda + 1 } { \mathrm { e } ^ { \lambda } } < S$ do
        $\lambda$ takes the value $\lambda + 1$
    End While
Output:
    Display $\lambda$
\end{verbatim}
a. What value does this algorithm display if we input the value $S = 0.8$?\\
b. What is the role of this algorithm?
\end{enumerate}