In the plane with an orthonormal coordinate system ($\mathrm { O } ; \vec { \imath } , \vec { \jmath }$), the representative curve $\mathscr { C }$ of a function $f$ defined and differentiable on the interval $] 0 ; + \infty [$ is given.

We have the following information:
\begin{itemize}
  \item the points $\mathrm { A } , \mathrm { B } , \mathrm { C }$ have coordinates $(1,0)$, $(1,2)$, $(0,2)$ respectively;
  \item the curve $\mathscr { C }$ passes through point B and the line (BC) is tangent to $\mathscr { C }$ at B;
  \item there exist two positive real numbers $a$ and $b$ such that for every strictly positive real number $x$,
$$f ( x ) = \frac { a + b \ln x } { x } .$$
\end{itemize}

\begin{enumerate}
  \item \begin{enumerate}
    \item[a.] Using the graph, give the values of $f ( 1 )$ and $f ^ { \prime } ( 1 )$.
    \item[b.] Verify that for every strictly positive real number $x , f ^ { \prime } ( x ) = \frac { ( b - a ) - b \ln x } { x ^ { 2 } }$.
    \item[c.] Deduce the real numbers $a$ and $b$.
  \end{enumerate}
  \item \begin{enumerate}
    \item[a.] Justify that for every real number $x$ in the interval $] 0 , + \infty [$, $f ^ { \prime } ( x )$ has the same sign as $- \ln x$.
    \item[b.] Determine the limits of $f$ at 0 and at $+ \infty$. We may note that for every strictly positive real number $x$, $f ( x ) = \frac { 2 } { x } + 2 \frac { \ln x } { x }$.
    \item[c.] Deduce the table of variations of the function $f$.
  \end{enumerate}
  \item \begin{enumerate}
    \item[a.] Prove that the equation $f ( x ) = 1$ has a unique solution $\alpha$ on the interval $] 0,1 ]$.
    \item[b.] By similar reasoning, we prove that there exists a unique real number $\beta$ in the interval $] 1 , + \infty [$ such that $f ( \beta ) = 1$. Determine the integer $n$ such that $n < \beta < n + 1$.
  \end{enumerate}
  \item The following algorithm is given.

\begin{center}
\begin{tabular}{|l|l|}
\hline
 & \begin{tabular}{l}
Variables: \\
$a , b$ and $m$ are real numbers. \\
Initialization: \\
Assign to $a$ the value 0. \\
Assign to $b$ the value 1. \\
Processing: \\
While $b - a > 0.1$ \\
Assign to $m$ the value $\frac { 1 } { 2 } ( a + b )$. \\
If $f ( m ) < 1$ then Assign to $a$ the value $m$. Otherwise Assign to $b$ the value $m$. \\
End If. \\
End While. \\
Output: \\
Display $a$. \\
Display $b$. \\
\end{tabular} \\
\hline
\end{tabular}
\end{center}

  \begin{enumerate}
    \item[a.] Run this algorithm by completing the table below, which you will copy onto your answer sheet.

\begin{center}
\begin{tabular}{ | c | c | c | l | l | l | }
\hline
 & step 1 & step 2 & step 3 & step 4 & step 5 \\
\hline
$a$ & 0 &  &  &  &  \\
\hline
$b$ & 1 &  &  &  &  \\
\hline
$b - a$ &  &  &  &  &  \\
\hline
$m$ &  &  &  &  &  \\
\hline
\end{tabular}
\end{center}

    \item[b.] What do the values displayed by this algorithm represent?
    \item[c.] Modify the algorithm above so that it displays the two bounds of an interval containing $\beta$ with amplitude $10 ^ { - 1 }$.
  \end{enumerate}
  \item The purpose of this question is to prove that the curve $\mathscr { C }$ divides the rectangle OABC into two regions of equal area.
  \begin{enumerate}
    \item[a.] Justify that this amounts to proving that $\int _ { \frac { 1 } { \mathrm { e } } } ^ { 1 } f ( x ) \mathrm { d } x = 1$.
    \item[b.] By noting that the expression of $f ( x )$ can be written as $\frac { 2 } { x } + 2 \times \frac { 1 } { x } \times \ln x$, complete the proof.
  \end{enumerate}
\end{enumerate}