In the plane with a coordinate system, we consider the curve $\mathscr{C}_{f}$ representative of a function $f$, twice differentiable on the interval $]0 ; +\infty[$.\\
The curve $\mathscr{C}_{f}$ admits a horizontal tangent line $T$ at point A(1;4).

\begin{enumerate}
  \item Specify the values $f(1)$ and $f^{\prime}(1)$.
\end{enumerate}

We admit that the function $f$ is defined for every real number $x$ in the interval $]0 ; +\infty[$ by:

$$f(x) = \frac{a + b \ln x}{x} \text{ where } a \text{ and } b \text{ are two real numbers.}$$

\begin{enumerate}
  \setcounter{enumi}{1}
  \item Prove that, for every strictly positive real number $x$, we have:
$$f^{\prime}(x) = \frac{b - a - b \ln x}{x^{2}}$$
  \item Deduce the values of the real numbers $a$ and $b$.
\end{enumerate}

In the rest of the exercise, we admit that the function $f$ is defined for every real number $x$ in the interval $]0; +\infty[$ by:

$$f(x) = \frac{4 + 4 \ln x}{x}$$

\begin{enumerate}
  \setcounter{enumi}{3}
  \item Determine the limits of $f$ at 0 and at $+\infty$.
  \item Determine the variation table of $f$ on the interval $]0 ; +\infty[$.
  \item Prove that, for every strictly positive real number $x$, we have:
$$f^{\prime\prime}(x) = \frac{-4 + 8 \ln x}{x^{3}}$$
  \item Show that the curve $\mathscr{C}_{f}$ has a unique inflection point B whose coordinates you will specify.
\end{enumerate}