We consider the function $f$ defined on $]0; 8]$ by
$$f ( x ) = \frac { 10 \ln \left( - x ^ { 2 } + 7 x + 9 \right) } { x }$$
Let $C _ { f }$ be the graphical representation of the function $f$ in an orthonormal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath})$.

\section*{Part A}
\begin{enumerate}
  \item Solve in $\mathbb { R }$ the inequality $- x ^ { 2 } + 7 x + 8 \geqslant 0$.
  \item Deduce that for all $x \in ] 0 ; 8 ]$, we have $f ( x ) \geqslant 0$.
  \item Interpret this result graphically.
\end{enumerate}

\section*{Part B}
The curve $C _ { f }$ is represented below.\\
Let $M$ be the point of $C _ { f }$ with abscissa $x$ where $x \in ] 0; 8]$.\\
We call $N$ and $P$ the orthogonal projections of the point $M$ respectively on the abscissa axis and on the ordinate axis.\\
In this part, we are interested in the area $\mathscr { A } ( x )$ of the rectangle $\mathrm{O}NMP$.

\begin{enumerate}
  \item Give the coordinates of points $N$ and $P$ as a function of $x$.
  \item Show that for all $x$ belonging to the interval $] 0 ; 8 ]$,
$$\mathscr { A } ( x ) = 10 \ln \left( - x ^ { 2 } + 7 x + 9 \right)$$
  \item Does there exist a position of the point $M$ for which the area of the rectangle $\mathrm{O}NMP$ is maximum? If it exists, determine this position.
\end{enumerate}

\section*{Part C}
We consider a strictly positive real number $k$.\\
We wish to determine the smallest value of $x$, approximated to the nearest tenth, belonging to $[ 3.5; 8 ]$ for which the area $\mathscr { A } ( x )$ becomes less than or equal to $k$.\\
To do this, we consider the algorithm below.\\
As a reminder, in Python language, $\ln ( x )$ is written log$(x)$.

\begin{verbatim}
from math import *
def A(x) :
    return 10*log (- 1* x**2 + 7*x + 9)
def pluspetitevaleur(k) :
    x = 3.5
    while A(x).........:
        x = x + 0.1
    return ...........
\end{verbatim}

\begin{enumerate}
  \item Copy and complete lines 8 and 10 of the algorithm.
  \item What number does the instruction \texttt{pluspetitevaleur(30)} then return?
  \item What happens when $k = 35$? Justify.
\end{enumerate}