\textbf{PART A}\\
We define on the interval $]0;+\infty[$ the function $g$ by:
$$g(x) = \frac{2}{x} - \frac{1}{x^2} + \ln x \text{ where ln denotes the natural logarithm function.}$$
We admit that the function $g$ is differentiable on $]0;+\infty[ = I$ and we denote by $g'$ its derivative function.
\begin{enumerate}
  \item Show that for $x > 0$, the sign of $g'(x)$ is that of the quadratic trinomial $(x^2 - 2x + 2)$.
  \item Deduce that the function $g$ is strictly increasing on $]0;+\infty[$.
  \item Show that the equation $g(x) = 0$ admits a unique solution on the interval $[0{,}5; 1]$, which we will denote $\alpha$.
  \item We are given the sign table of $g$ on the interval $]0;+\infty[ = I$:
\begin{center}
\begin{tabular}{ c || l l l l }
$x$ & 0 & & $\alpha$ & $+\infty$ \\
\hline
$g(x)$ & $-$ & 0 & $+$ & \\
\hline
\end{tabular}
\end{center}
Justify this sign table using the results obtained in the previous questions.
\end{enumerate}

\textbf{PART B}\\
We consider the function $f$ defined on the interval $]0;+\infty[ = I$ by:
$$f(x) = \mathrm{e}^x \ln x.$$
We denote by $\mathscr{C}_f$ the representative curve of $f$ in an orthonormal coordinate system.
\begin{enumerate}
  \item We admit that the function $f$ is twice differentiable on $]0;+\infty[$, we denote by $f'$ its derivative function, $f''$ its second derivative function and we admit that: for every real number $x > 0$, $f'(x) = \mathrm{e}^x\left(\frac{1}{x} + \ln x\right)$.\\
  Prove that, for every real number $x > 0$, we have: $f''(x) = \mathrm{e}^x\left(\frac{2}{x} - \frac{1}{x^2} + \ln x\right)$.
  \item We may note that for every real $x > 0$, $f''(x) = \mathrm{e}^x \times g(x)$, where $g$ denotes the function studied in part A.
  \item a. Draw the sign table of the function $f''$ on $]0;+\infty[$. Justify.\\
  b. Justify that the curve $\mathscr{C}_f$ admits a unique inflection point A.\\
  c. Study the convexity of the function $f$ on the interval $]0;+\infty[$. Justify.
  \item a. Calculate the limits of $f$ at the boundaries of its domain of definition.\\
  b. Show that $f'(\alpha) = \frac{\mathrm{e}^\alpha}{\alpha^2}(1-\alpha)$.\\
  We recall that $\alpha$ is the unique solution of the equation $g(x) = 0$.\\
  c. Prove that $f'(\alpha) > 0$ and deduce the sign of $f'(x)$ for $x$ belonging to $]0;+\infty[$.\\
  d. Deduce the complete variation table of the function $f$ on $]0;+\infty[$.
\end{enumerate}