\textbf{Main topics covered: Exponential function; differentiation; convexity}

\section*{Part 1}
Below is given, in the plane referred to an orthonormal reference frame, the curve representing the derivative function $f'$ of a function $f$ differentiable on $\mathbb{R}$.\\
Using this curve, conjecture, by justifying the answers:
\begin{enumerate}
  \item The direction of variation of the function $f$ on $\mathbb{R}$.
  \item The convexity of the function $f$ on $\mathbb{R}$.
\end{enumerate}

\section*{Part 2}
It is admitted that the function $f$ mentioned in Part 1 is defined on $\mathbb{R}$ by:
$$f(x) = (x+2)\mathrm{e}^{-x}.$$
We denote $\mathscr{C}$ the representative curve of $f$ in an orthonormal reference frame $(\mathrm{O}; \vec{\imath}, \vec{\jmath})$.\\
It is admitted that the function $f$ is twice differentiable on $\mathbb{R}$, and we denote $f'$ and $f''$ the first and second derivative functions of $f$ respectively.

\begin{enumerate}
  \item Show that, for every real number $x$,
$$f(x) = \frac{x}{\mathrm{e}^x} + 2\mathrm{e}^{-x}.$$
Deduce the limit of $f$ at $+\infty$.\\
Justify that the curve $\mathscr{C}$ admits an asymptote which you will specify.\\
It is admitted that $\lim_{x \rightarrow -\infty} f(x) = -\infty$.
  \item a. Show that, for every real number $x$, $f'(x) = (-x-1)\mathrm{e}^{-x}$.\\
b. Study the variations on $\mathbb{R}$ of the function $f$ and draw up its variation table.\\
c. Show that the equation $f(x) = 2$ admits a unique solution $\alpha$ on the interval $[-2;-1]$ and give an approximate value to the nearest $10^{-1}$.
  \item Determine, for every real number $x$, the expression of $f''(x)$ and study the convexity of the function $f$.
\end{enumerate}

What does point A with abscissa 0 represent for the curve $\mathscr{C}$?