We consider, for every integer $n > 0$, the functions $f_{n}$ defined on the interval $[1; 5]$ by:
$$f_{n}(x) = \frac{\ln x}{x^{n}}$$
For every integer $n > 0$, we denote by $\mathscr{C}_{n}$ the representative curve of the function $f_{n}$ in an orthogonal reference frame.

\begin{enumerate}
  \item Show that, for every integer $n > 0$ and every real $x$ in the interval $[1; 5]$:
$$f_{n}^{\prime}(x) = \frac{1 - n\ln(x)}{x^{n+1}}$$
  \item For every integer $n > 0$, we admit that the function $f_{n}$ has a maximum on the interval $[1; 5]$. We denote by $A_{n}$ the point of the curve $\mathscr{C}_{n}$ having as ordinate this maximum.\\
Show that all points $A_{n}$ belong to the same curve $\Gamma$ with equation
$$y = \frac{1}{\mathrm{e}} \ln(x)$$
  \item a. Show that, for every integer $n > 1$ and every real $x$ in the interval $[1; 5]$:
$$0 \leqslant \frac{\ln(x)}{x^{n}} \leqslant \frac{\ln(5)}{x^{n}}$$
b. Show that for every integer $n > 1$:
$$\int_{1}^{5} \frac{1}{x^{n}} \mathrm{~d}x = \frac{1}{n-1}\left(1 - \frac{1}{5^{n-1}}\right)$$
c. For every integer $n > 0$, we are interested in the area, expressed in square units, of the surface under the curve $f_{n}$, that is the area of the region of the plane bounded by the lines with equations $x = 1$, $x = 5$, $y = 0$ and the curve $\mathscr{C}_{n}$.\\
Determine the limiting value of this area as $n$ tends to $+\infty$.
\end{enumerate}