Let $f$ be a function defined and differentiable on $\mathbb{R}$. We denote by $\mathscr{C}$ its representative curve in the plane equipped with a coordinate system $(\mathrm{O}, \vec{\imath}, \vec{\jmath})$.

\textbf{Part A}

In the graphs below, we have represented the curve $\mathscr{C}$ and three other curves $\mathscr{C}_1, \mathscr{C}_2, \mathscr{C}_3$ with the tangent line at their point with abscissa 0.

\begin{enumerate}
  \item Determine by reading the graph, the sign of $f(x)$ according to the values of $x$.
  \item We denote by $F$ a primitive of the function $f$ on $\mathbb{R}$.\\
a. Using the curve $\mathscr{C}$, determine $F'(0)$ and $F'(-2)$.\\
b. One of the curves $\mathscr{C}_1, \mathscr{C}_2, \mathscr{C}_3$ is the representative curve of the function $F$. Determine which one by justifying the elimination of the other two.
\end{enumerate}

\textbf{Part B}

In this part, we assume that the function $f$ mentioned in Part A is the function defined on $\mathbb{R}$ by
$$f(x) = (x+2)\mathrm{e}^{\frac{1}{2}x}$$

\begin{enumerate}
  \item Observation of the curve $\mathscr{C}$ allows us to conjecture that the function $f$ admits a minimum.\\
a. Prove that for all real $x$, $f'(x) = \frac{1}{2}(x+4)\mathrm{e}^{\frac{1}{2}x}$.\\
b. Deduce a validation of the previous conjecture.
  \item Let $I = \int_0^1 f(x)\,\mathrm{d}x$.\\
a. Give a geometric interpretation of the real number $I$.\\
b. Let $u$ and $v$ be the functions defined on $\mathbb{R}$ by $u(x) = x$ and $v(x) = \mathrm{e}^{\frac{1}{2}x}$. Verify that $f = 2(u'v + uv')$.\\
c. Deduce the exact value of the integral $I$.
  \item The algorithm below is given.

\begin{center}
\begin{tabular}{|l|l|}
\hline
Variables : & $k$ and $n$ are natural integers. $s$ is a real number. \\
\hline
Input : & Assign to $s$ the value 0. \\
\hline
Processing : & For $k$ ranging from 0 to $n-1$ \\
\hline
 & End of loop. \\
\hline
Output : & Display $s$. \\
\hline
\end{tabular}
\end{center}

We denote by $s_n$ the number displayed by this algorithm when the user enters a strictly positive natural integer as the value of $n$.\\
a. Justify that $s_3$ represents the area, expressed in square units, of the shaded region in the graph below where the three rectangles have the same width.\\
b. What can be said about the value of $s_n$ provided by the proposed algorithm when $n$ becomes large?
\end{enumerate}