\textbf{Exercise 3 Functions, logarithm function}

Let $g$ be the function defined on the interval $]0; +\infty[$ by
$$g(x) = 1 + x^{2}[1 - 2\ln(x)]$$
The function $g$ is differentiable on the interval $]0; +\infty[$ and we denote $g'$ its derivative function.\\
We call $\mathscr{C}$ the representative curve of the function $g$ in an orthonormal coordinate system of the plane.

\section*{PART A}
\begin{enumerate}
  \item Justify that $g(\mathrm{e})$ is strictly negative.
  \item Justify that $\lim_{x \rightarrow +\infty} g(x) = -\infty$.
  \item a. Show that, for all $x$ belonging to the interval $]0; +\infty[$, $g'(x) = -4x\ln(x)$.\\
b. Study the direction of variation of the function $g$ on the interval $]0; +\infty[$.\\
c. Show that the equation $g(x) = 0$ admits a unique solution, denoted $\alpha$, on the interval $[1; +\infty[$.\\
d. Give an interval for $\alpha$ with amplitude $10^{-2}$.
  \item Deduce from the above the sign of the function $g$ on the interval $[1; +\infty[$.
\end{enumerate}

\section*{PART B}
\begin{enumerate}
  \item We admit that, for all $x$ belonging to the interval $[1; \alpha]$, $g''(x) = -4[\ln(x) + 1]$. Justify that the function $g$ is concave on the interval $[1; \alpha]$.
  \item In the figure opposite, A and B are points on the curve $\mathscr{C}$ with abscissae respectively 1 and $\alpha$.\\
a. Determine the reduced equation of the line (AB).\\
b. Deduce from this that for all real $x$ belonging to the interval $[1; \alpha]$,
$$g(x) \geqslant \frac{-2}{\alpha - 1} x + \frac{2\alpha}{\alpha - 1}.$$
\end{enumerate}