Main topics covered: Logarithm function, limits, differentiation.

\section*{Part 1}
The graph below gives the graphical representation in an orthonormal coordinate system of the function $f$ defined on the interval $]0; +\infty[$ by:

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

\begin{enumerate}
  \item Determine by calculation the unique solution $\alpha$ of the equation $f(x) = 0$.

Give the exact value of $\alpha$ as well as the value rounded to the nearest hundredth.\\
  \item Specify, by graphical reading, the sign of $f(x)$ when $x$ varies in the interval $]0; +\infty[$.
\end{enumerate}

\section*{Part II}
We consider the function $g$ defined on the interval $]0; +\infty[$ by:

$$g(x) = [\ln(x)]^2 - \ln(x)$$

\begin{enumerate}
  \item a. Determine the limit of the function $g$ at 0.\\
b. Determine the limit of the function $g$ at $+\infty$.
  \item We denote by $g'$ the derivative function of the function $g$ on the interval $]0; +\infty[$.

Prove that, for any real number $x$ in $]0; +\infty[$, we have: $g'(x) = f(x)$, where $f$ denotes the function defined in Part I.
  \item Draw up the variation table of the function $g$ on the interval $]0; +\infty[$.

This table should include the limits of the function $g$ at 0 and at $+\infty$, as well as the value of the minimum of $g$ on $]0; +\infty[$.
  \item Prove that, for any real number $m > -0.25$, the equation $g(x) = m$ has exactly two solutions.
  \item Determine by calculation the two solutions of the equation $g(x) = 0$.
\end{enumerate}