We consider $n$ a non-zero natural integer. We consider the function $f_n$ defined on the interval $[0; 1]$ by:

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

We admit that the function $f_n$ is differentiable on $[0; 1]$ and we denote $f_n'$ its derivative function.

\section*{Part A}
In this part we study the case where $n = 1$. We thus study the function $f_1$ defined on $[0; 1]$ by:

$$f_1(x) = x e^{1-x}$$

\begin{enumerate}
  \item Show that $f_1'(x)$ is strictly positive for all real $x$ in $[0; 1[$.
  \item Deduce the table of variations of the function $f_1$ on the interval $[0; 1]$.
  \item Deduce that the equation $f_1(x) = 0.1$ admits a unique solution in the interval $[0; 1]$.
\end{enumerate}

\section*{Part B}
We consider the sequence $(u_n)$ defined for all non-zero natural integers $n$ by

$$u_n = \int_0^1 f_n(x) \, dx \quad \text{that is} \quad u_n = \int_0^1 x^n e^{1-x} \, dx$$

We admit that $u_1 = e - 2$.

\begin{enumerate}
  \item a. Justify that for all $x \in [0; 1]$ and for all non-zero natural integers $n$,
$$0 \leq x^{n+1} \leq x^n$$
  b. Deduce that for all non-zero natural integers $n$,
$$0 \leq u_{n+1} \leq u_n.$$
  c. Show that the sequence $(u_n)$ is convergent.
  \item a. Using integration by parts, prove that for all non-zero natural integers $n$ we have:
$$u_{n+1} = (n+1)u_n - 1$$
  b. Consider the Python script below defining the function suite():
\begin{verbatim}
from math import exp
def suite():
    u = ...
    for n in range (1, ...):
        u = ...
    return
\end{verbatim}
Copy and complete the Python script above so that the function suite() returns the value of $\int_0^1 x^8 e^{1-x} \, dx$.
  \item a. Prove that for all non-zero natural integers $n$ we have:
$$u_n \leq \frac{e}{n+1}$$
  b. Deduce the limit of the sequence $(u_n)$.
\end{enumerate}