\section*{Exercise 3 — 7 points}
Topics: Logarithm function, Sequences\\
\section*{Parts $\mathbf { B }$ and $\mathbf { C }$ are independent}
We consider the function $f$ defined on $] 0 ; + \infty [$ by
$$f ( x ) = x - x \ln x ,$$
where ln denotes the natural logarithm function.

\section*{Part A}
\begin{enumerate}
  \item Determine the limit of $f ( x )$ as $x$ tends to 0.
  \item Determine the limit of $f ( x )$ as $x$ tends to $+ \infty$.
  \item We admit that the function $f$ is differentiable on $] 0 ; + \infty \left[ \right.$ and we denote by $f ^ { \prime }$ its derivative function.\\
a. Prove that, for every real number $x > 0$, we have: $f ^ { \prime } ( x ) = - \ln x$.\\
b. Deduce the variations of the function $f$ on $] 0 ; + \infty [$ and draw its variation table.
  \item Solve the equation $f ( x ) = x$ on $] 0$; $+ \infty [$.
\end{enumerate}

\section*{Part B}
In this part, you may use with profit certain results from Part A.\\
We consider the sequence $(u _ { n })$ defined by:
$$\begin{cases} u _ { 0 } & = 0.5 \\ u _ { n + 1 } & = u _ { n } - u _ { n } \ln u _ { n } \text { for every natural number } n , \end{cases}$$
Thus, for every natural number $n$, we have: $u _ { n + 1 } = f \left( u _ { n } \right)$.
\begin{enumerate}
  \item We recall that the function $f$ is increasing on the interval $[ 0.5 ; 1 ]$.\\
Prove by induction that, for every natural number $n$, we have: $0.5 \leqslant u _ { n } \leqslant u _ { n + 1 } \leqslant 1$.
  \item a. Show that the sequence $( u _ { n } )$ is convergent.\\
b. We denote by $\ell$ the limit of the sequence $( u _ { n } )$. Determine the value of $\ell$.
\end{enumerate}

\section*{Part C}
For any real number $k$, we consider the function $f _ { k }$ defined on $] 0 ; + \infty [$ by:
$$f _ { k } ( x ) = k x - x \ln x$$
\begin{enumerate}
  \item For every real number $k$, show that $f _ { k }$ admits a maximum $y _ { k }$ attained at $x _ { k } = \mathrm { e } ^ { k - 1 }$.
  \item Verify that, for every real number $k$, we have: $x _ { k } = y _ { k }$.
\end{enumerate}