\section*{Exercise 1 -- Common to all candidates}

\section*{Part A}
Let $f$ be the function defined on $\mathbb{R}$ by
$$f(x) = \frac{3}{1 + \mathrm{e}^{-2x}}$$
In the graph below, we have drawn, in an orthogonal coordinate system $(\mathrm{O}, \vec{\imath}, \vec{\jmath})$, the representative curve $\mathscr{C}$ of the function $f$ and the line $\Delta$ with equation $y = 3$.

\begin{enumerate}
  \item Prove that the function $f$ is strictly increasing on $\mathbb{R}$.
  \item Justify that the line $\Delta$ is an asymptote to the curve $\mathscr{C}$.
  \item Prove that the equation $f(x) = 2.999$ has a unique solution $\alpha$ on $\mathbb{R}$.
\end{enumerate}
Determine an interval containing $\alpha$ with amplitude $10^{-2}$.

\section*{Part B}
Let $h$ be the function defined on $\mathbb{R}$ by $h(x) = 3 - f(x)$.
\begin{enumerate}
  \item Justify that the function $h$ is positive on $\mathbb{R}$.
  \item We denote by $H$ the function defined on $\mathbb{R}$ by $H(x) = -\frac{3}{2}\ln\left(1 + \mathrm{e}^{-2x}\right)$.

Prove that $H$ is an antiderivative of $h$ on $\mathbb{R}$.
  \item Let $a$ be a strictly positive real number.\\
  a. Give a graphical interpretation of the integral $\int_{0}^{a} h(x)\,\mathrm{d}x$.\\
  b. Prove that $\int_{0}^{a} h(x)\,\mathrm{d}x = \frac{3}{2}\ln\left(\frac{2}{1 + \mathrm{e}^{-2a}}\right)$.\\
  c. We denote by $\mathscr{D}$ the set of points $M(x\,;\,y)$ in the plane defined by
$$\left\{\begin{array}{l} x \geqslant 0 \\ f(x) \leqslant y \leqslant 3 \end{array}\right.$$
Determine the area, in square units, of the region $\mathscr{D}$.
\end{enumerate}