\textbf{Part A}

Below, in an orthogonal coordinate system, are the curves $\mathscr{C}_1$ and $\mathscr{C}_2$, graphical representations of two functions defined and differentiable on $\mathbb{R}$. One of the two functions represented is the derivative of the other. We will denote them $g$ and $g'$.\\
We also specify that:
\begin{itemize}
  \item The curve $\mathscr{C}_1$ intersects the y-axis at the point with coordinates $(0; 1)$.
  \item The curve $\mathscr{C}_2$ intersects the y-axis at the point with coordinates $(0; 2)$ and the x-axis at the points with coordinates $(-2; 0)$ and $(1; 0)$.
\end{itemize}

\begin{enumerate}
  \item By justifying, associate to each of the functions $g$ and $g'$ its graphical representation.
  \item Justify that the equation of the tangent line to the curve representing the function $g$ at the point with x-coordinate 0 is $y = 2x + 1$.
\end{enumerate}

\textbf{Part B}

We consider $(E)$ the differential equation
$$y + y' = (2x + 3)\mathrm{e}^{-x}$$
where $y$ is a function of the real variable $x$.

\begin{enumerate}
  \item Show that the function $f_0$ defined for every real number $x$ by $f_0(x) = (x^2 + 3x)\mathrm{e}^{-x}$ is a particular solution of the differential equation $(E)$.
  \item Solve the differential equation $(E_0): y + y' = 0$.
  \item Determine the solutions of the differential equation $(E)$.
  \item We admit that the function $g$ described in Part A is a solution of the differential equation $(E)$. Then determine the expression of the function $g$.
  \item Determine the solutions of the differential equation $(E)$ whose curve has exactly two inflection points.
\end{enumerate}

\textbf{Part C}

We consider the function $f$ defined for every real number $x$ by:
$$f(x) = (x^2 + 3x + 2)\mathrm{e}^{-x}$$

\begin{enumerate}
  \item Prove that the limit of the function $f$ at $+\infty$ is equal to 0.

We also admit that the limit of the function $f$ at $-\infty$ is equal to $+\infty$.
  \item We admit that the function $f$ is differentiable on $\mathbb{R}$. We denote by $f'$ the derivative function of $f$ on $\mathbb{R}$.\\
a. Verify that, for every real number $x$, $f'(x) = (-x^2 - x + 1)\mathrm{e}^{-x}$.\\
b. Determine the sign of the derivative function $f'$ on $\mathbb{R}$ and then deduce the variations of the function $f$ on $\mathbb{R}$.
  \item Explain why the function $f$ is positive on the interval $[0; +\infty[$.
  \item We will denote by $\mathscr{C}_f$ the curve representing the function $f$ in an orthogonal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath})$. We admit that the function $F$ defined for every real number $x$ by $F(x) = (-x^2 - 5x - 7)\mathrm{e}^{-x}$ is a primitive of the function $f$.\\
Let $\alpha$ be a positive real number.\\
Determine the area $\mathscr{A}(\alpha)$, expressed in square units, of the region of the plane bounded by the x-axis, the curve $\mathscr{C}_f$ and the lines with equations $x = 0$ and $x = \alpha$.
\end{enumerate}