4. Aviation regulations stipulate that two aircraft on approach must be at least 3 nautical miles apart at all times (1 nautical mile equals 1852 m). If aircraft Alpha and Beta are respectively at $E$ and $F$ at the same instant, is their safety distance respected?

\section*{Exercise 3. (5 points)}
The plane is equipped with an orthonormal coordinate system.\\
For every natural number $n$, we consider the function $f_n$ defined on $[0; +\infty[$ by:

$$f_0(x) = \mathrm{e}^{-x} \text{ and, for } n \geq 1, f_n(x) = x^n \mathrm{e}^{-x}.$$

For every natural number $n$, we denote $C_n$ the representative curve of function $f_n$.\\
Parts A and B are independent.

\section*{Part A: Study of functions $\boldsymbol{f}_{\boldsymbol{n}}$ for $\boldsymbol{n} \geq \mathbf{1}$}
We consider a natural number $n \geq 1$.

\begin{enumerate}
  \item a. We admit that function $f_n$ is differentiable on $[0; +\infty[$.
\end{enumerate}

Show that for all $x \geq 0$,

$$f_n'(x) = (n - x)x^{n-1}\mathrm{e}^{-x}$$

b. Justify all elements of the table below:

\begin{center}
\begin{tabular}{|c|ccccc|}
\hline
$x$ & 0 &  & $n$ &  & $+\infty$ \\
\hline
$f_n'(x)$ &  & + & 0 & - &  \\
\hline
 &  &  & $\left(\frac{n}{\mathrm{e}}\right)^n$ &  &  \\
$f(x)$ &  &  &  &  &  \\
 & 0 &  &  & 0 &  \\
\hline
\end{tabular}
\end{center}

\begin{enumerate}
  \setcounter{enumi}{1}
  \item Justify by calculation that point $A\left(1; \mathrm{e}^{-1}\right)$ belongs to curve $C_n$.
\end{enumerate}

\section*{Part B: Study of integrals $\int_0^1 \boldsymbol{f}_{\boldsymbol{n}}(\boldsymbol{x})\mathrm{d}\boldsymbol{x}$ for $\boldsymbol{n} \geq \mathbf{0}$}
In this part, we study functions $f_n$ on $[0; 1]$ and we consider the sequence $(I_n)$ defined for every natural number $n$ by:

$$I_n = \int_0^1 f_n(x)\mathrm{d}x = \int_0^1 x^n \mathrm{e}^{-x}\mathrm{d}x$$

\begin{enumerate}
  \item On the graph in APPENDIX (page 9/9), curves $C_0, C_1, C_2, C_{10}$ and $C_{100}$ are represented.\\
a. Give a graphical interpretation of $I_n$.\\
b. By reading this graph, what conjecture can be made about the limit of sequence $(I_n)$?
  \item Calculate $I_0$.
  \item a. Let $n$ be a natural number.
\end{enumerate}

Prove that for all $x \in [0; 1]$,

$$0 \leq x^{n+1} \leq x^n.$$

b. Deduce that for every natural number $n$, we have:

$$0 \leq I_{n+1} \leq I_n$$

\begin{enumerate}
  \setcounter{enumi}{3}
  \item Prove that sequence $(I_n)$ is convergent, towards a limit greater than or equal to zero that we denote $\ell$.
  \item Using integration by parts, prove that for every natural number $n$ we have:
\end{enumerate}

$$I_{n+1} = (n+1)I_n - \frac{1}{\mathrm{e}}$$

\begin{enumerate}
  \setcounter{enumi}{5}
  \item a. Prove that if $\ell > 0$, the equality from question 5 leads to a contradiction.\\
b. Prove that $\ell = 0$. You may use question 6.a.
\end{enumerate}

Below is the script of the mystere function, written in Python. The constant e has been imported.

\begin{verbatim}
def mystere(n):
    I = 1 - 1/e
    L = [I]
    for i in range(n):
        I = (i + 1)*I - 1/e
        L.append(I)
    return L
\end{verbatim}

\begin{enumerate}
  \setcounter{enumi}{6}
  \item What does mystere(100) return in the context of the exercise?
\end{enumerate}

\section*{Exercise 4. (5 points)}
For each of the following statements, indicate whether it is true or false. Each answer must be justified. An unjustified answer earns no points. In this exercise, the questions are independent of one another.

\begin{enumerate}
  \item We consider the differential equation (E):
\end{enumerate}

$$y' = \frac{1}{2}y + 4.$$

Statement 1: The solutions of (E) are the functions $f$ defined on $\mathbb{R}$ by:

$$f(x) = k\mathrm{e}^{\frac{1}{2}x} - 8, \text{ with } k \in \mathbb{R}$$

\begin{enumerate}
  \setcounter{enumi}{1}
  \item In a final year class, there are 18 girls and 14 boys.
\end{enumerate}

A volleyball team is formed by randomly choosing 3 girls and 3 boys.\\
Statement 2: There are 297024 possibilities for forming such a team.\\