\textbf{EXERCISE B - Differential equation}

\textbf{Part A: Determination of a function $f$ and resolution of a differential equation}

Consider the function $f$ defined on $\mathbb{R}$ by:
$$f(x) = \mathrm{e}^x + ax + b\mathrm{e}^{-x}$$
where $a$ and $b$ are real numbers that we propose to determine in this part. In the plane with a coordinate system with origin O, the curve $\mathscr{C}$, representing the function $f$, and the tangent line $(T)$ to the curve $\mathscr{C}$ at the point with abscissa 0 are shown.

\begin{enumerate}
  \item By reading the graph, give the values of $f(0)$ and $f'(0)$.
  \item Using the expression of the function $f$, express $f(0)$ as a function of $b$ and deduce the value of $b$.
  \item We admit that the function $f$ is differentiable on $\mathbb{R}$ and we denote by $f'$ its derivative function.\\
  a. Give, for every real $x$, the expression of $f'(x)$.\\
  b. Express $f'(0)$ as a function of $a$.\\
  c. Using the previous questions, determine $a$, then deduce the expression of $f(x)$.
  \item Consider the differential equation:
$$( E ) : \quad y' + y = 2\mathrm{e}^x - x - 1$$
  a. Verify that the function $g$ defined on $\mathbb{R}$ by:
$$g(x) = \mathrm{e}^x - x + 2\mathrm{e}^{-x}$$
is a solution of equation $(E)$.\\
  b. Solve the differential equation $y' + y = 0$.\\
  c. Deduce all solutions of equation $(E)$.
\end{enumerate}

\textbf{Part B: Study of the function $g$ on $[1;+\infty[$}

\begin{enumerate}
  \item Verify that for every real $x$, we have:
$$\mathrm{e}^{2x} - \mathrm{e}^x - 2 = (\mathrm{e}^x - 2)(\mathrm{e}^x + 1)$$
  \item Deduce a factored expression of $g'(x)$, for every real $x$.
  \item We will admit that, for all $x \in [1;+\infty[$, $\mathrm{e}^x - 2 > 0$.\\
  Study the direction of variation of the function $g$ on $[1;+\infty[$.
\end{enumerate}