6. During the national blood donation week, a blood collection is organized in $N$ French cities chosen at random numbered $1,2,3 , \ldots , N$ where $N$ is a non-zero natural integer.\\
We consider the random variable $X _ { 1 }$ which associates to each sample of 100 people from city 1 the number of universal donors in that sample.\\
We define in the same way the random variables $X _ { 2 }$ for city $2 , \ldots , X _ { N }$ for city $N$.

We assume that these random variables are independent and that they have the same expectation equal to 7.14 and the same variance equal to 6.63. We consider the random variable $M _ { N } = \frac { X _ { 1 } + X _ { 2 } + \cdots + X _ { N } } { N }$.\\
a. What does the random variable $M _ { N }$ represent in the context of the exercise?\\
b. Calculate the expectation $E \left( M _ { N } \right)$.\\
c. We denote by $V \left( M _ { N } \right)$ the variance of the random variable $M _ { N }$.

Show that $V \left( M _ { N } \right) = \frac { 6,63 } { N }$.\\
d. Determine the smallest value of $N$ for which Bienaymé-Chebyshev's inequality allows us to assert that:

$$P \left( 7 < M _ { N } < 7,28 \right) \geq 0,95 .$$

\section*{Exercise 2 (6 points)}
We consider a function $f$ defined on the interval $] 0 ; + \infty [$. We admit that it is twice differentiable on the interval $] 0 ; + \infty \left[ \right.$. We denote by $f ^ { \prime }$ its derivative function and $f ^ { \prime \prime }$ its second derivative function.

In an orthogonal coordinate system, we have drawn below:

\begin{itemize}
  \item the representative curve of $f$, denoted $C _ { f }$, on the interval $\left. ] 0 ; 3 \right]$;
  \item the line $T _ { A }$, tangent to $C _ { f }$ at point $A ( 1 ; 2 )$;
  \item the line $T _ { B }$, tangent to $C _ { f }$ at point $B ( \mathrm { e } ; \mathrm { e } )$.
\end{itemize}

We further specify that the tangent $T _ { A }$ passes through point $C ( 3 ; 0 )$.\\
\includegraphics[max width=\textwidth, alt={}, center]{d067dc4c-cc79-48c1-860a-1bd730c54967-4_1225_1536_861_267}

\section*{Part A: Graphical readings}
We will answer the following questions by justifying them using the graph.

\begin{enumerate}
  \item Determine the derivative number $f ^ { \prime } ( 1 )$.
  \item How many solutions does the equation $f ^ { \prime } ( x ) = 0$ have in the interval $] 0$; 3 ] ?
  \item What is the sign of $f ^ { \prime \prime } ( 0,2 )$ ?
\end{enumerate}

\section*{Part B: Study of the function $\boldsymbol { f }$}
We admit in this part that the function $f$ is defined on the interval $] 0 ; + \infty [$ by:

$$f ( x ) = x \left( 2 ( \ln x ) ^ { 2 } - 3 \ln x + 2 \right)$$

where ln denotes the natural logarithm function.

\begin{enumerate}
  \item Solve in $\mathbb { R }$ the equation $2 X ^ { 2 } - 3 X + 2 = 0$.
\end{enumerate}

Deduce that $C _ { f }$ does not intersect the x-axis.\\