Throughout the exercise, $n$ denotes a strictly positive natural number. The purpose of the exercise is to study the equation

$$\left( E _ { n } \right) : \quad \frac { \ln ( x ) } { x } = \frac { 1 } { n }$$

with unknown strictly positive real number $x$.

\section*{Part A}
Let $f$ be the function defined on the interval $] 0$; $+ \infty [$ by

$$f ( x ) = \frac { \ln ( x ) } { x }$$

It is admitted that the function $f$ is differentiable on the interval $] 0 ; + \infty [$.

\begin{enumerate}
  \item Study the variations of function $f$.
  \item Determine its maximum.
\end{enumerate}

\section*{Part B}
\begin{enumerate}
  \item Show that, for $n \geqslant 3$, the equation $f ( x ) = \frac { 1 } { n }$ has a unique solution on $[ 1 ; e]$ denoted $\alpha _ { n }$.
  \item From the above, for every integer $n \geqslant 3$, the real number $\alpha _ { n }$ is a solution of equation $\left( E _ { n } \right)$.\\
a. On the graph are drawn the lines $D _ { 3 } , D _ { 4 }$ and $D _ { 5 }$ with equations respectively $y = \frac { 1 } { 3 } , y = \frac { 1 } { 4 }$, $y = \frac { 1 } { 5 }$.\\
Conjecture the direction of variation of the sequence ( $\alpha _ { n }$ ).\\
b. Compare, for every integer $n \geqslant 3 , f \left( \alpha _ { n } \right)$ and $f \left( \alpha _ { n + 1 } \right)$.\\
Determine the direction of variation of the sequence $\left( \alpha _ { n } \right)$.\\
c. Deduce that the sequence ( $\alpha _ { n }$ ) converges.\\
It is not asked to calculate its limit.
  \item It is admitted that, for every integer $n \geqslant 3$, equation $\left( E _ { n } \right)$ has another solution $\beta _ { n }$ such that
$$1 \leqslant \alpha _ { n } \leqslant \mathrm { e } \leqslant \beta _ { n }$$
a. It is admitted that the sequence ( $\beta _ { n }$ ) is increasing.\\
Establish that, for every natural number $n$ greater than or equal to 3,
$$\beta _ { n } \geqslant n \frac { \beta _ { 3 } } { 3 } .$$
b. Deduce the limit of the sequence ( $\beta _ { n }$ ).
\end{enumerate}