We consider the function $g$ defined for all real $x$ in the interval $[0;1]$ by:
$$g(x) = 1 + \mathrm{e}^{-x}$$
We admit that, for all real $x$ in the interval $[0;1], g(x) > 0$.

We denote $\mathscr{C}$ the representative curve of function $g$ in an orthogonal coordinate system, and $\mathscr{D}$ the plane region bounded on one hand between the $x$-axis and curve $\mathscr{C}$, on the other hand between the lines with equations $x = 0$ and $x = 1$.

The purpose of this exercise is to divide region $\mathscr{D}$ into two regions of equal area, first by a line parallel to the $y$-axis (part A), then by a line parallel to the $x$-axis (part B).

\section*{Part A}
Let $a$ be a real number such that $0 \leqslant a \leqslant 1$.\\
We denote $\mathscr{A}_1$ the area of the region between curve $\mathscr{C}$, the $x$-axis, the lines with equations $x = 0$ and $x = a$, and $\mathscr{A}_2$ that of the region between curve $\mathscr{C}$, the $x$-axis and the lines with equations $x = a$ and $x = 1$.\\
$\mathscr{A}_1$ and $\mathscr{A}_2$ are expressed in square units.

\begin{enumerate}
  \item a. Prove that $\mathscr{A}_1 = a - \mathrm{e}^{-a} + 1$.\\
  b. Express $\mathscr{A}_2$ as a function of $a$.
  \item Let $f$ be the function defined for all real $x$ in the interval $[0;1]$ by:
$$f(x) = 2x - 2\mathrm{e}^{-x} + \frac{1}{\mathrm{e}}$$
  a. Draw the variation table of function $f$ on the interval $[0;1]$. The exact values of $f(0)$ and $f(1)$ will be specified.\\
  b. Prove that function $f$ vanishes once and only once on the interval $[0;1]$, at a real number $\alpha$. Give the value of $\alpha$ rounded to the nearest hundredth.
  \item Using the previous questions, determine an approximate value of the real $a$ for which the areas $\mathscr{A}_1$ and $\mathscr{A}_2$ are equal.
\end{enumerate}

\section*{Part B}
Let $b$ be a positive real number.\\
In this part, we propose to divide region $\mathscr{D}$ into two regions of equal area by the line with equation $y = b$. We admit that there exists a unique positive real $b$ that is a solution.

\begin{enumerate}
  \item Justify the inequality $b < 1 + \frac{1}{\mathrm{e}}$. You may use a graphical argument.
  \item Determine the exact value of the real $b$.
\end{enumerate}