We consider the function $f$ defined on the interval $]0; +\infty[$ by
$$f(x) = (2 - \ln x) \times \ln x,$$
where ln denotes the natural logarithm function.

We admit that the function $f$ is twice differentiable on $]0; +\infty[$.

We denote by $C$ the representative curve of the function $f$ in an orthogonal coordinate system and $C'$ the representative curve of the function $f'$, the derivative function of the function $f$.

The curve $\boldsymbol{C}'$ is given (with its unique horizontal tangent (T)).

\begin{enumerate}
  \item By graphical reading, with the precision that the above diagram allows, give:\\
  a. the slope of the tangent to $C$ at the point with abscissa 1.\\
  b. the largest interval on which the function $f$ is convex.
  \item a. Calculate the limit of the function $f$ at $+\infty$.\\
  b. Calculate $\lim_{x \rightarrow 0} f(x)$. Interpret this result graphically.
  \item Show that the curve $C$ intersects the x-axis at exactly two points, whose coordinates you will specify.
  \item a. Show that for all real $x$ belonging to $]0; +\infty[$, $f'(x) = \dfrac{2(1 - \ln x)}{x}$.\\
  b. Deduce, by justifying, the table of variations of the function $f$ on $]0; +\infty[$.
  \item We denote by $f''$ the second derivative of $f$ and we admit that for all real $x$ belonging to $]0; +\infty[$, $f''(x) = \dfrac{2(\ln x - 2)}{x^2}$.\\
  Determine by calculation the largest interval on which the function $f$ is convex and specify the coordinates of the inflection point of the curve $C$.
\end{enumerate}