\section*{Part A: study of the function $\boldsymbol { f }$}
The function $f$ is defined on the interval $] 0$; $+ \infty$ [ by:

$$f ( x ) = x - 2 + \frac { 1 } { 2 } \ln x$$

where ln denotes the natural logarithm function. We admit that the function $f$ is twice differentiable on $] 0 ; + \infty \left[ \right.$, we denote by $f ^ { \prime }$ its derivative and $f ^ { \prime \prime }$ its second derivative.

\begin{enumerate}
  \item a. Determine, by justifying, the limits of $f$ at 0 and at $+ \infty$.\\
b. Show that for all $x$ belonging to $] 0$; $+ \infty \left[ \right.$, we have: $f ^ { \prime } ( x ) = \frac { 2 x + 1 } { 2 x }$.\\
c. Study the direction of variation of $f$ on $] 0 ; + \infty [$.\\
d. Study the convexity of $f$ on $] 0 ; + \infty [$.
  \item a. Show that the equation $f ( x ) = 0$ admits in $] 0 ; + \infty [$ a unique solution which we denote by $\alpha$ and justify that $\alpha$ belongs to the interval $[ 1 ; 2 ]$.\\
b. Determine the sign of $f ( x )$ for $x \in ] 0$; $+ \infty [$.\\
c. Show that $\ln ( \alpha ) = 2 ( 2 - \alpha )$.
\end{enumerate}

\section*{Part B: study of the function $g$}
The function $g$ is defined on $] 0 ; 1]$ by:

$$g ( x ) = - \frac { 7 } { 8 } x ^ { 2 } + x - \frac { 1 } { 4 } x ^ { 2 } \ln x .$$

We admit that the function $g$ is differentiable on $] 0 ; 1 ]$ and we denote by $g ^ { \prime }$ its derivative function.

\begin{enumerate}
  \item Calculate $g ^ { \prime } ( x )$ for $\left. x \in \right] 0$; 1] then verify that $g ^ { \prime } ( x ) = x f \left( \frac { 1 } { x } \right)$.
  \item a. Justify that for $x$ belonging to the interval $] 0$; $\frac { 1 } { \alpha } \left[ \right.$, we have $f \left( \frac { 1 } { x } \right) > 0$.\\
b. We admit the following sign table:
\end{enumerate}

\begin{center}
\begin{tabular}{ | c | | c c c c | }
\hline
$x$ & \multicolumn{1}{|c}{0} & $\frac { 1 } { \alpha }$ & 1 &  \\
\hline
sign of $f \left( \frac { 1 } { x } \right)$ &  & + & 0 & - \\
\hline
\end{tabular}
\end{center}

Deduce the variation table of $g$ on the interval $] 0 ; 1 ]$. Images and limits are not required.

\section*{Part C: an area calculation}
The following are represented on the graph below:

\begin{itemize}
  \item The curve $\mathscr { C } _ { g }$ of the function $g$;
  \item The parabola $\mathscr { P }$ with equation $y = - \frac { 7 } { 8 } x ^ { 2 } + x$ on the interval $\left. ] 0 ; 1 \right]$.
\end{itemize}

We wish to calculate the area $\mathscr { A }$ of the shaded region between the curves $\mathscr { C } _ { g }$ and $\mathscr { P }$, and the lines with equations $x = \frac { 1 } { \alpha }$ and $x = 1$.\\
We recall that $\ln ( \alpha ) = 2 ( 2 - \alpha )$.

\begin{enumerate}
  \item a. Justify the relative position of the curves $C _ { g }$ and $\mathscr { P }$ on the interval $\left. ] 0 ; 1 \right]$.\\
b. Prove the equality:
$$\int _ { \frac { 1 } { \alpha } } ^ { 1 } x ^ { 2 } \ln x \mathrm {~d} x = \frac { - \alpha ^ { 3 } - 6 \alpha + 13 } { 9 \alpha ^ { 3 } }$$
  \item Deduce the expression as a function of $\alpha$ of the area $\mathscr { A }$.
\end{enumerate}