\textbf{Main topics covered: Natural logarithm function; convexity}

We consider the function $f$ defined on the interval $]0;+\infty[$ by:
$$f(x) = x + 4 - 4\ln(x) - \frac{3}{x}$$
where ln denotes the natural logarithm function.\\
We denote $\mathscr{C}$ the graphical representation of $f$ in an orthonormal coordinate system.

\begin{enumerate}
  \item Determine the limit of the function $f$ at $+\infty$.
  \item We assume that the function $f$ is differentiable on $]0;+\infty[$ and we denote $f^{\prime}$ its derivative function.

Prove that, for every real number $x > 0$, we have:
$$f^{\prime}(x) = \frac{x^{2} - 4x + 3}{x^{2}}$$
  \item a. Give the variation table of the function $f$ on the interval $]0;+\infty[$.

The exact values of the extrema and the limits of $f$ at 0 and at $+\infty$ will be shown. We will assume that $\lim_{x \rightarrow 0} f(x) = -\infty$.\\
b. By simply reading the variation table, specify the number of solutions of the equation $f(x) = \frac{5}{3}$.
  \item Study the convexity of the function $f$, that is, specify the parts of the interval $]0;+\infty[$ on which $f$ is convex, and those on which $f$ is concave.\\
We will justify that the curve $\mathscr{C}$ admits a unique inflection point, whose coordinates we will specify.
\end{enumerate}