2. Let $Y$ denote the random variable which, for a small jar randomly selected from the production of line $\mathrm { F } _ { 2 }$, associates its sugar content.\\
We assume that $Y$ follows the normal distribution with mean $m _ { 2 } = 0.17$ and standard deviation $\sigma _ { 2 }$.\\
We further assume that the probability that a small jar randomly selected from the production of line $\mathrm { F } _ { 2 }$ is compliant equals 0.99 .\\
Let Z be the random variable defined by $Z = \frac { Y - m _ { 2 } } { \sigma _ { 2 } }$.\\
a. What distribution does the random variable $Z$ follow?\\
b. Determine, as a function of $\sigma _ { 2 }$, the interval to which $Z$ belongs when $Y$ belongs to the interval $[ 0.16 ; 0.18 ]$.\\
c. Deduce an approximate value to $10 ^ { - 3 }$ near of $\sigma _ { 2 }$.

You may use the table given below, in which the random variable $Z$ follows the normal distribution with mean 0 and standard deviation 1 .

\begin{center}
\begin{tabular}{ | c | c | }
\hline
$\beta$ & $P ( - \beta \leqslant Z \leqslant \beta )$ \\
\hline
2.4324 & 0.985 \\
\hline
2.4573 & 0.986 \\
\hline
2.4838 & 0.987 \\
\hline
2.5121 & 0.988 \\
\hline
2.5427 & 0.989 \\
\hline
2.5758 & 0.990 \\
\hline
2.6121 & 0.991 \\
\hline
2.6521 & 0.992 \\
\hline
2.6968 & 0.993 \\
\hline
\end{tabular}
\end{center}

\section*{Exercise 3}
\section*{Common to all candidates}
Given a real number $k$, we consider the function $f _ { k }$ defined on $\mathbb { R }$ by

$$f _ { k } ( x ) = \frac { 1 } { 1 + \mathrm { e } ^ { - k x } }$$

The plane is equipped with an orthonormal coordinate system ( $\mathrm { O } , \vec { \imath } , \vec { \jmath }$ ).

\section*{Part A}
In this part we choose $k = 1$. We have therefore, for all real $x , f _ { 1 } ( x ) = \frac { 1 } { 1 + \mathrm { e } ^ { - x } }$.\\
The graph $\mathscr { C } _ { 1 }$ of the function $f _ { 1 }$ in the coordinate system ( $\mathrm { O } , \vec { \imath } , \vec { \jmath }$ ) is given in APPENDIX, to be returned with your answer sheet.

\begin{enumerate}
  \item Determine the limits of $f _ { 1 } ( x )$ as $x \to + \infty$ and as $x \to - \infty$ and give a graphical interpretation of the results obtained.
  \item Prove that, for all real $x , f _ { 1 } ( x ) = \frac { \mathrm { e } ^ { x } } { 1 + \mathrm { e } ^ { x } }$.
  \item Let $f _ { 1 } ^ { \prime }$ denote the derivative function of $f _ { 1 }$ on $\mathbb { R }$. Calculate, for all real $x , f _ { 1 } ^ { \prime } ( x )$.
\end{enumerate}

Deduce the variations of the function $f _ { 1 }$ on $\mathbb { R }$.\\