\textbf{Part A}\\
Consider the function $f$ defined on $\mathbb{R}$ by
$$f(x) = \left(x + \frac{1}{2}\right)e^{-x} + x.$$
\begin{enumerate}
  \item Determine the limits of $f$ at $-\infty$ and at $+\infty$.
  \item It is admitted that $f$ is twice differentiable on $\mathbb{R}$.\\
  a. Prove that, for all $x \in \mathbb{R}$,
  $$f''(x) = \left(x - \frac{3}{2}\right)e^{-x}.$$
  b. Deduce the variations and the minimum of the function $f'$ on $\mathbb{R}$.\\
  c. Justify that for all $x \in \mathbb{R}$, $f'(x) > 0$.\\
  d. Deduce that the equation $f(x) = 0$ admits a unique solution on $\mathbb{R}$.\\
  e. Give a value rounded to $10^{-3}$ of this solution.
\end{enumerate}

\textbf{Part B}\\
Consider a function $h$, defined and differentiable on $\mathbb{R}$, having an expression of the form
$$h(x) = (ax + b)e^{-x} + x \text{, where } a \text{ and } b \text{ are two real numbers.}$$
In an orthonormal coordinate system shown below are:
\begin{itemize}
  \item the representative curve $\mathscr{C}_h$ of the function $h$;
  \item the points A and B with coordinates respectively $(-2; -2.5)$ and $(2; 3.5)$.
\end{itemize}
\begin{enumerate}
  \item Conjecture, with the precision allowed by the graph, the abscissae of any inflection points of the representative curve of the function $h$.
  \item Knowing that the function $h$ admits on $\mathbb{R}$ a second derivative with expression
  $$h''(x) = -\frac{3}{2}e^{-x} + xe^{-x}.$$
  validate or not the previous conjecture.
  \item Determine an equation of the line (AB).
  \item Knowing that the line (AB) is tangent to the representative curve of the function $h$ at the point with abscissa 0, deduce the values of $a$ and $b$.
\end{enumerate}