Consider the function $f$ defined on $] 0 ; + \infty [$ by

$$f ( x ) = x ^ { 2 } - x \ln ( x ) .$$

We admit that $f$ is twice differentiable on $] 0 ; + \infty [$.
We denote $f ^ { \prime }$ the derivative function of $f$ and $f ^ { \prime \prime }$ the derivative function of $f ^ { \prime }$.

\textbf{Part A: Study of the function $f$}

\begin{enumerate}
  \item Determine the limits of the function $f$ at 0 and at $+ \infty$.
  \item For all strictly positive real $x$, calculate $f ^ { \prime } ( x )$.
  \item Show that for all strictly positive real $x$:
$$f ^ { \prime \prime } ( x ) = \frac { 2 x - 1 } { x }$$
  \item Study the variations of the function $f ^ { \prime }$ on $] 0 ; + \infty [$, then draw up the table of variations of the function $f ^ { \prime }$ on $] 0 ; + \infty [$.\\
Care should be taken to show the exact value of the extremum of the function $f ^ { \prime }$ on $] 0 ; + \infty [$.\\
The limits of the function $f ^ { \prime }$ at the boundaries of the domain of definition are not expected.
  \item Show that the function $f$ is strictly increasing on $] 0 ; + \infty [$.
\end{enumerate}

\textbf{Part B: Study of an auxiliary function for solving the equation $f ( x ) = x$}

We consider in this part the function $g$ defined on $] 0 ; + \infty [$ by

$$g ( x ) = x - \ln ( x )$$

We admit that the function $g$ is differentiable on $] 0 ; + \infty [$, we denote $g ^ { \prime }$ its derivative.

\begin{enumerate}
  \item For all strictly positive real, calculate $g ^ { \prime } ( x )$, then draw up the table of variations of the function $g$.\\
The limits of the function $g$ at the boundaries of the domain of definition are not expected.
  \item We admit that 1 is the unique solution of the equation $g ( x ) = 1$. Solve, on the interval $] 0 ; + \infty [$, the equation $f ( x ) = x$.
\end{enumerate}

\textbf{Part C: Study of a recursive sequence}

We consider the sequence $( u _ { n } )$ defined by $u _ { 0 } = \frac { 1 } { 2 }$ and for all natural integer $n$,

$$u _ { n + 1 } = f \left( u _ { n } \right) = u _ { n } ^ { 2 } - u _ { n } \ln \left( u _ { n } \right) .$$

\begin{enumerate}
  \item Show by induction that for all natural integer $n$:
$$\frac { 1 } { 2 } \leqslant u _ { n } \leqslant u _ { n + 1 } \leqslant 1 .$$
  \item Justify that the sequence $( u _ { n } )$ converges.\\
We call $\ell$ the limit of the sequence $( u _ { n } )$ and we admit that $\ell$ satisfies the equality $f ( \ell ) = \ell$.
  \item Determine the value of $\ell$.
\end{enumerate}