\textbf{Exercise 2 -- Common to all candidates}

Consider the functions $f$ and $g$ defined for all real $x$ by:
$$f(x) = \mathrm{e}^{x} \quad \text{and} \quad g(x) = 1 - \mathrm{e}^{-x}.$$
The representative curves of these functions in an orthogonal coordinate system of the plane, denoted respectively $\mathscr{C}_{f}$ and $\mathscr{C}_{g}$, are provided in the appendix.

\textbf{Part A}

These curves appear to admit two common tangent lines. Draw these tangent lines as accurately as possible on the figure in the appendix.

\textbf{Part B}

In this part, the existence of these common tangent lines is admitted.\\
Let $\mathscr{D}$ denote one of them. This line is tangent to the curve $\mathscr{C}_{f}$ at point A with abscissa $a$ and tangent to the curve $\mathscr{C}_{g}$ at point B with abscissa $b$.

\begin{enumerate}
  \item a. Express in terms of $a$ the slope of the tangent line to the curve $\mathscr{C}_{f}$ at point A.\\
b. Express in terms of $b$ the slope of the tangent line to the curve $\mathscr{C}_{g}$ at point B.\\
c. Deduce that $b = -a$.
  \item Prove that the real number $a$ is a solution of the equation
$$2(x - 1)\mathrm{e}^{x} + 1 = 0.$$
\end{enumerate}

\textbf{Part C}

Consider the function $\varphi$ defined on $\mathbb{R}$ by
$$\varphi(x) = 2(x - 1)\mathrm{e}^{x} + 1$$

\begin{enumerate}
  \item a. Calculate the limits of the function $\varphi$ at $-\infty$ and $+\infty$.\\
b. Calculate the derivative of the function $\varphi$, then study its sign.\\
c. Draw the variation table of the function $\varphi$ on $\mathbb{R}$. Specify the value of $\varphi(0)$.
  \item a. Prove that the equation $\varphi(x) = 0$ has exactly two solutions in $\mathbb{R}$.\\
b. Let $\alpha$ denote the negative solution of the equation $\varphi(x) = 0$ and $\beta$ the positive solution of this equation.\\
Using a calculator, give the values of $\alpha$ and $\beta$ rounded to the nearest hundredth.
\end{enumerate}

\textbf{Part D}

In this part, we prove the existence of these common tangent lines, which was admitted in Part B.\\
Let E be the point on the curve $\mathscr{C}_{f}$ with abscissa $\alpha$ and F the point on the curve $\mathscr{C}_{g}$ with abscissa $-\alpha$ ($\alpha$ is the real number defined in Part C).

\begin{enumerate}
  \item Prove that the line $(EF)$ is tangent to the curve $\mathscr{C}_{f}$ at point E.
  \item Prove that $(EF)$ is tangent to $\mathscr{C}_{g}$ at point F.
\end{enumerate}