\textbf{Exercise 3 (7 points)}

\textbf{Part 1}

Let $g$ be the function defined for every real number $x$ in the interval $]0; +\infty[$ by:
$$g(x) = \frac{2\ln x}{x}$$

\begin{enumerate}
  \item Let $g'$ denote the derivative of $g$. Prove that for every strictly positive real $x$:
$$g'(x) = \frac{2 - 2\ln x}{x^2}$$
  \item We have the following variation table for the function $g$ on the interval $]0; +\infty[$:
\begin{center}
\begin{tabular}{ | c | | | | c c c c | }
\hline
$x$ & 0 & 1 & e & $+\infty$ \\
\hline
\begin{tabular}{ c }
Variations \\
of $g$ \\
\end{tabular} & & & ${}^{\frac{2}{\mathrm{e}}}$ & \\
 & & & & \\
 & & & & \\
\hline
\end{tabular}
\end{center}
Justify the following information read from this table:\\
a. the value $\frac{2}{\mathrm{e}}$;\\
b. the variations of the function $g$ on its domain;\\
c. the limits of the function $g$ at the boundaries of its domain.
  \item Deduce the sign table of the function $g$ on the interval $]0; +\infty[$.
\end{enumerate}

\textbf{Part 2}

Let $f$ be the function defined on the interval $]0; +\infty[$ by
$$f(x) = [\ln(x)]^2.$$
In this part, each study is carried out on the interval $]0; +\infty[$.

\begin{enumerate}
  \item Prove that on the interval $]0; +\infty[$, the function $f$ is a primitive of the function $g$.
  \item Using Part 1, study:\\
  a. the convexity of the function $f$;\\
  b. the variations of the function $f$.
  \item a. Give an equation of the tangent line to the curve representing the function $f$ at the point with abscissa $e$.\\
  b. Deduce that, for every real $x$ in $]0; e]$:
$$[\ln(x)]^2 \geqslant \frac{2}{\mathrm{e}} x - 1$$
\end{enumerate}