Let $f$ be the function defined on the interval $] 0 ; + \infty [$ by

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

and let $\mathscr { C }$ be the representative curve of the function $f$ in a coordinate system of the plane.

\begin{enumerate}
  \item a. Study the limit of $f$ at 0.\\
b. What is $\lim _ { x \rightarrow + \infty } \frac { \ln ( x ) } { x }$ ? Deduce the limit of the function $f$ at $+ \infty$.\\
c. Deduce the possible asymptotes to the curve $\mathscr { C }$.
  \item a. Let $f ^ { \prime }$ denote the derivative function of the function $f$ on the interval $] 0 ; + \infty [$. Prove that, for every real $x$ belonging to the interval $] 0 ; + \infty [$,
$$f ^ { \prime } ( x ) = \frac { - 1 - 2 \ln ( x ) } { x ^ { 3 } }$$
b. Solve on the interval $] 0 ; + \infty [$ the inequality $- 1 - 2 \ln ( x ) > 0$.\\
Deduce the sign of $f ^ { \prime } ( x )$ on the interval $] 0 ; + \infty [$.\\
c. Draw up the table of variations of the function $f$.
  \item a. Prove that the curve $\mathscr { C }$ has a unique point of intersection with the $x$-axis, whose coordinates you will specify.\\
b. Deduce the sign of $f ( x )$ on the interval $] 0 ; + \infty [$.
  \item For every integer $n \geqslant 1$, we denote by $I _ { n }$ the area, expressed in square units, of the region bounded by the $x$-axis, the curve $\mathscr { C }$ and the lines $x = 1$ and $x = n$.
\end{enumerate}