\textbf{Exercise 2: Functions, logarithm function}

The purpose of this exercise is to study the function $f$, defined on $]0; +\infty[$, by:
$$f(x) = 3x - x\ln(x) - 2\ln(x).$$

\textbf{PART A: Study of an auxiliary function $g$}

Let $g$ be the function defined on $]0; +\infty[$ by
$$g(x) = 2(x-1) - x\ln(x)$$
We denote $g'$ the derivative function of $g$. We admit that $\lim_{x \rightarrow +\infty} g(x) = -\infty$.
\begin{enumerate}
  \item Calculate $g(1)$ and $g(\mathrm{e})$.
  \item Determine $\lim_{x \rightarrow 0^+} g(x)$ by justifying your approach.
  \item Show that, for all $x > 0$, $g'(x) = 1 - \ln(x)$.\\
    Deduce the variation table of $g$ on $]0; +\infty[$.
  \item Show that the equation $g(x) = 0$ has exactly two distinct solutions on $]0; +\infty[$: 1 and $\alpha$ with $\alpha$ belonging to the interval $[\mathrm{e}; +\infty[$.\\
    Give an approximation of $\alpha$ to 0.01.
  \item Deduce the sign table of $g$ on $]0; +\infty[$.
\end{enumerate}

\textbf{PART B: Study of the function $f$}

We consider in this part the function $f$, defined on $]0; +\infty[$, by
$$f(x) = 3x - x\ln(x) - 2\ln(x).$$
We denote $f'$ the derivative function of $f$.\\
We admit that: $\lim_{x \rightarrow 0^+} f(x) = +\infty$.
\begin{enumerate}
  \item Determine the limit of $f$ at $+\infty$ by justifying your approach.
  \item a. Justify that for all $x > 0$, $f'(x) = \dfrac{g(x)}{x}$.\\
    b. Deduce the variation table of $f$ on $]0; +\infty[$.
  \item We admit that, for all $x > 0$, the second derivative of $f$, denoted $f''$, is defined by $f''(x) = \dfrac{2-x}{x^2}$.\\
    Study the convexity of $f$ and specify the coordinates of the inflection point of $\mathscr{C}_f$.
\end{enumerate}