We consider a function $f$ defined on the interval $] 0 ; + \infty [$. We admit that it is twice differentiable on the interval $] 0 ; + \infty[$. We denote by $f ^ { \prime }$ its derivative function and $f ^ { \prime \prime }$ its second derivative function.

In an orthogonal coordinate system, we have drawn:
\begin{itemize}
  \item the representative curve of $f$, denoted $\mathscr { C } _ { f }$ on the interval $] 0$; 3 ];
  \item the line $T _ { \mathrm { A } }$, tangent to $\mathscr { C } _ { f }$ at point $\mathrm { A } ( 1 ; 2 )$;
  \item the line $T _ { \mathrm { B } }$ tangent to $\mathscr { C } _ { f }$ at point $\mathrm { B } ( \mathrm { e } ; \mathrm { e } )$.
\end{itemize}
We further specify that the tangent $T _ { \mathrm { A } }$ passes through point $\mathrm { C } ( 3 ; 0 )$.

\textbf{Part A: Graphical readings}

Answer the following questions by justifying them using the graph.
\begin{enumerate}
  \item Determine the derivative number $f ^ { \prime } ( 1 )$.
  \item How many solutions does the equation $f ^ { \prime } ( x ) = 0$ have in the interval $] 0$; 3 ]?
  \item What is the sign of $f ^ { \prime \prime } ( 0{,}2 )$ ?
\end{enumerate}

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

We admit in this part that the function $f$ is defined on the interval $] 0 ; + \infty [$ by
$$f ( x ) = x \left[ 2 ( \ln x ) ^ { 2 } - 3 \ln x + 2 \right]$$
where ln denotes the natural logarithm function.

\begin{enumerate}
  \item Solve in $\mathbb { R }$ the equation $2 X ^ { 2 } - 3 X + 2 = 0$. Deduce that $\mathscr { C } _ { f }$ does not intersect the x-axis.
  \item Determine, by justifying, the limit of $f$ as $+ \infty$. We will admit that the limit of $f$ at 0 is equal to 0.
  \item We admit that for all $x$ belonging to $] 0 ; + \infty [$, $f ^ { \prime } ( x ) = 2 ( \ln x ) ^ { 2 } + \ln x - 1$.\\
  a. Show that for all $x$ belonging to $] 0 ; + \infty [$, $f ^ { \prime \prime } ( x ) = \frac { 1 } { x } ( 4 \ln x + 1 )$.\\
  b. Study the convexity of the function $f$ on the interval $] 0 ; + \infty [$ and specify the exact value of the abscissa of the inflection point.\\
  c. Show that the curve $\mathscr { C } _ { f }$ is above the tangent $T _ { \mathrm { B } }$ on the interval $] 1 ; + \infty [$.
\end{enumerate}

\textbf{Part C: Area calculation}

\begin{enumerate}
  \item Justify that the tangent $T _ { \mathrm { B } }$ has the reduced equation $y = 2 x - \mathrm { e }$.
  \item Using integration by parts, show that
  $$\int _ { 1 } ^ { \mathrm { e } } x \ln x \mathrm {~d} x = \frac { \mathrm { e } ^ { 2 } + 1 } { 4 }$$
  \item We denote by $\mathscr { A }$ the area of the shaded region in the figure, bounded by the curve $\mathscr { C } _ { f }$, the tangent $T _ { \mathrm { B } }$, and the lines with equations $x = 1$ and $x = \mathrm { e }$. We admit that $\int _ { 1 } ^ { \mathrm { e } } x ( \ln x ) ^ { 2 } \mathrm {~d} x = \frac { \mathrm { e } ^ { 2 } - 1 } { 4 }$. Deduce the exact value of $\mathscr { A }$ in square units.
\end{enumerate}