\textbf{Exercise 2 — 7 points}\\
Themes: Logarithm function and sequence\\
Let $f$ be the function defined on the interval $]0;+\infty[$ by
$$f(x) = x\ln(x) + 1$$
We denote $\mathscr{C}_f$ its representative curve in a coordinate system of the plane.

\begin{enumerate}
  \item Determine the limit of the function $f$ at $0$ as well as its limit at $+\infty$.

  \item \begin{enumerate}
    \item[a.] We admit that $f$ is differentiable on $]0;+\infty[$ and we denote $f'$ its derivative function. Show that for every strictly positive real number $x$:
$$f'(x) = 1 + \ln(x).$$
    \item[b.] Deduce the variation table of the function $f$ on $]0;+\infty[$. The exact value of the extremum of $f$ on $]0;+\infty[$ and the limits must be shown.
    \item[c.] Justify that for all $x \in ]0;1[$, $f(x) \in ]0;1[$.
  \end{enumerate}

  \item \begin{enumerate}
    \item[a.] Determine an equation of the tangent line $(T)$ to the curve $\mathscr{C}_f$ at the point with abscissa $1$.
    \item[b.] Study the convexity of the function $f$ on $]0;+\infty[$.
    \item[c.] Deduce that for every strictly positive real number $x$:
$$f(x) \geqslant x$$
  \end{enumerate}

  \item The sequence $(u_n)$ is defined by its first term $u_0$ element of the interval $]0;1[$ and for every natural number $n$:
$$u_{n+1} = f(u_n)$$
  \begin{enumerate}
    \item[a.] Prove by induction that for every natural number $n$, we have: $0 < u_n < 1$.
    \item[b.] Deduce from question 3.c. the increasing nature of the sequence $(u_n)$.
    \item[c.] Deduce that the sequence $(u_n)$ is convergent.
  \end{enumerate}
\end{enumerate}