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

\section*{Part 1}
Let $h$ denote the function defined on the interval $]0; +\infty[$ by:
$$h(x) = 1 + \frac{\ln(x)}{x^2}.$$
It is admitted that the function $h$ is differentiable on $]0; +\infty[$ and we denote $h'$ its derivative function.

\begin{enumerate}
  \item Determine the limits of $h$ at 0 and at $+\infty$.
  \item Show that, for every real number $x$ in $]0; +\infty[$, $h'(x) = \frac{1 - 2\ln(x)}{x^3}$.
  \item Deduce the variations of the function $h$ on the interval $]0; +\infty[$.
  \item Show that the equation $h(x) = 0$ admits a unique solution $\alpha$ belonging to $]0; +\infty[$ and verify that: $\frac{1}{2} < \alpha < 1$.
  \item Determine the sign of $h(x)$ for $x$ belonging to $]0; +\infty[$.
\end{enumerate}

\section*{Part 2}
Let $f_1$ and $f_2$ denote the functions defined on $]0; +\infty[$ by:
$$f_1(x) = x - 1 - \frac{\ln(x)}{x^2} \quad \text{and} \quad f_2(x) = x - 2 - \frac{2\ln(x)}{x^2}.$$
We denote $\mathscr{C}_1$ and $\mathscr{C}_2$ the respective graphs of $f_1$ and $f_2$ in a reference frame $(\mathrm{O}; \vec{\imath}, \vec{\jmath})$.

\begin{enumerate}
  \item Show that, for every real number $x$ belonging to $]0; +\infty[$, we have:
$$f_1(x) - f_2(x) = h(x).$$
  \setcounter{enumi}{1}
  \item Deduce from the results of Part 1 the relative position of the curves $\mathscr{C}_1$ and $\mathscr{C}_2$.
\end{enumerate}
You will justify that their unique point of intersection has coordinates $(\alpha; \alpha)$.\\
Recall that $\alpha$ is the unique solution of the equation $h(x) = 0$.