\textbf{Main topics covered: Logarithm function; differentiation.}

\textbf{Part I: Study of an auxiliary function}

Let $g$ be the function defined on $]0; +\infty[$ by:

$$g(x) = \ln(x) + 2x - 2.$$

\begin{enumerate}
  \item Determine the limits of $g$ at $+\infty$ and 0.
  \item Determine the direction of variation of the function $g$ on $]0; +\infty[$.
  \item Prove that the equation $g(x) = 0$ admits a unique solution $\alpha$ on $]0; +\infty[$.
  \item Calculate $g(1)$ then determine the sign of $g$ on $]0; +\infty[$.
\end{enumerate}

\textbf{Part II: Study of a function $f$}

We consider the function $f$, defined on $]0; +\infty[$ by:

$$f(x) = \left(2 - \frac{1}{x}\right)(\ln(x) - 1)$$

\begin{enumerate}
  \item a. We assume that the function $f$ is differentiable on $]0; +\infty[$ and we denote by $f^{\prime}$ its derivative. Prove that, for every $x$ in $]0; +\infty[$, we have:
$$f^{\prime}(x) = \frac{g(x)}{x^{2}}$$
b. Draw the variation table of the function $f$ on $]0; +\infty[$. The calculation of limits is not required.
  \item Solve the equation $f(x) = 0$ on $]0; +\infty[$ then draw the sign table of $f$ on the interval $]0; +\infty[$.
\end{enumerate}

\textbf{Part III: Study of a function $F$ whose derivative is the function $f$}

We assume that there exists a function $F$ differentiable on $]0; +\infty[$ whose derivative $F^{\prime}$ is the function $f$. Thus, we have: $F^{\prime} = f$.\\
We denote by $\mathscr{C}_{F}$ the representative curve of the function $F$ in an orthonormal coordinate system $(\mathrm{O}, \vec{\imath}, \vec{\jmath})$. We will not seek to determine an expression for $F(x)$.

\begin{enumerate}
  \item Study the variations of $F$ on $]0; +\infty[$.
  \item Does the curve $\mathscr{C}_{F}$ representative of $F$ admit tangent lines parallel to the x-axis? Justify the answer.
\end{enumerate}