\section*{Exercise 3 — 6 points}
Theme: Exponential function\\
Main topics covered: Sequences; Functions, Exponential function.

\section*{Part A}
We consider the function $f$ defined for every real $x$ by:
$$f ( x ) = 1 + x - \mathrm { e } ^ { 0,5 x - 2 } .$$
We assume that the function $f$ is differentiable on $\mathbb { R }$. We denote $f ^ { \prime }$ its derivative.

\begin{enumerate}
  \item a. Determine the limit of the function $f$ at $- \infty$.\\
b. Prove that, for every non-zero real $x$, $f ( x ) = 1 + 0,5 x \left( 2 - \frac { \mathrm { e } ^ { 0,5 x } } { 0,5 x } \times \mathrm { e } ^ { - 2 } \right)$.\\
Deduce the limit of the function $f$ at $+ \infty$.
  \item a. Determine $f ^ { \prime } ( x )$ for every real $x$.\\
b. Prove that the set of solutions of the inequality $f ^ { \prime } ( x ) < 0$ is the interval $] 4 + 2 \ln ( 2 ) ; + \infty [$.
  \item Deduce from the previous questions the variation table of the function $f$ on $\mathbb { R }$. The exact value of the image of $4 + 2 \ln ( 2 )$ by $f$ should be shown.
  \item Show that the equation $f ( x ) = 0$ has a unique solution on the interval $[ - 1 ; 0 ]$.
\end{enumerate}

\section*{Part B}
We consider the sequence $( u _ { n } )$ defined by $u _ { 0 } = 0$ and, for every natural number $n$, $u _ { n + 1 } = f \left( u _ { n } \right)$ where $f$ is the function defined in Part A.

\begin{enumerate}
  \item a. Prove by induction that, for every natural number $n$, we have:
$$u _ { n } \leqslant u _ { n + 1 } \leqslant 4 .$$
b. Deduce that the sequence $( u _ { n } )$ converges. We denote its limit by $\ell$.
  \item a. We recall that $f$ satisfies the relation $\ell = f ( \ell )$. Prove that $\ell = 4$.\\
b. We consider the function value written below in the Python language:
\begin{verbatim}
def valeur (a) :
    u = 0
    n = 0
    while u <= a:
        u=1 + u - exp(0.5*u - 2)
        n = n+1
    return n
\end{verbatim}
The instruction valeur(3.99) returns the value 12.\\
Interpret this result in the context of the exercise.
\end{enumerate}