The purpose of this exercise is to study the function $f$ defined on the interval $]0; +\infty[$ by:

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

\textbf{Part A: graphical readings}

Below is the representative curve $(\mathscr{C}_f)$ of the function $f$, as well as the line $(T)$, tangent to the curve $(\mathscr{C}_f)$ at point A with coordinates $(1; -1)$. This tangent also passes through the point $B(0; -4)$.

\begin{enumerate}
  \item Read graphically $f'(1)$ and give the reduced equation of the tangent $(T)$.
  \item Give the intervals on which the function $f$ appears to be convex or concave. What does point A appear to represent for the curve $(\mathscr{C}_f)$?
\end{enumerate}

\textbf{Part B: analytical study}

\begin{enumerate}
  \item Determine, by justifying, the limit of $f$ at $+\infty$, then its limit at 0.
  \item It is admitted that the function $f$ is twice differentiable on the interval $]0; +\infty[$.\\
a. Determine $f'(x)$ for $x$ belonging to the interval $]0; +\infty[$.\\
b. Show that for all $x$ belonging to the interval $]0; +\infty[$,
$$f''(x) = \frac{2(x+1)(x-1)}{x^3}.$$
  \item a. Study the convexity of the function $f$ on the interval $]0; +\infty[$.\\
b. Study the variations of the function $f'$, then the sign of $f'(x)$ for $x$ belonging to the interval $]0; +\infty[$.\\
Deduce the direction of variation of the function $f$ on the interval $]0; +\infty[$.
  \item a. Show that the equation $f(x) = 0$ has a unique solution $\alpha$ on the interval $]0; +\infty[$.\\
b. Give the value of $\alpha$ rounded to the nearest hundredth and show that $\alpha$ satisfies:
$$\alpha^2 = \exp\left(\frac{1}{\alpha^2}\right)$$
\end{enumerate}