\textbf{Exercise 2} — Main topics covered: functions, logarithm function.

Let $f$ be the function defined on the interval $]0; +\infty[$ by:
$$f(x) = x\ln(x) - x - 2.$$
We admit that the function $f$ is twice differentiable on $]0; +\infty[$. We denote $f'$ its derivative, $f''$ its second derivative and $\mathcal{C}_f$ its representative curve in a coordinate system.

\begin{enumerate}
  \item a. Prove that, for all $x$ belonging to $]0; +\infty[$, we have $f'(x) = \ln(x)$.\\
  b. Determine an equation of the tangent line $T$ to the curve $\mathcal{C}_f$ at the point with abscissa $x = \mathrm{e}$.\\
  c. Justify that the function $f$ is convex on the interval $]0; +\infty[$.\\
  d. Deduce the relative position of the curve $\mathcal{C}_f$ and the tangent line $T$.

  \item a. Calculate the limit of the function $f$ at 0.\\
  b. Prove that the limit of the function $f$ at $+\infty$ is equal to $+\infty$.

  \item Draw up the table of variations of the function $f$ on the interval $]0; +\infty[$.

  \item a. Prove that the equation $f(x) = 0$ has a unique solution in the interval $]0; +\infty[$. We denote this solution by $\alpha$.\\
  b. Justify that the real number $\alpha$ belongs to the interval $]4{,}3; 4{,}4[$.\\
  c. Deduce the sign of the function $f$ on the interval $]0; +\infty[$.

  \item Consider the following threshold function written in Python:\\
  Recall that the \texttt{log} function of the \texttt{math} module (which we assume is imported) denotes the natural logarithm function $\ln$.
\begin{verbatim}
def seuil(pas) :
    x=4.3
    while x*log(x) - x - 2 < 0:
        x=x+pas
    return x
\end{verbatim}
  What is the value returned when calling the function \texttt{seuil(0.01)}? Interpret this result in the context of the exercise.
\end{enumerate}