For each of the following statements, indicate whether it is true or false.\\
Each answer must be justified. An unjustified answer receives no points.
\begin{enumerate}
  \item Consider the function $f$ defined on $\mathbb{R}$ by
$$f(x) = \mathrm{e}^{x} + x.$$
Statement A: The function $f$ has the following variation table:
\begin{center}
\begin{tabular}{|l|l|}
\hline
$x$ & $-\infty$ \\
\hline
variations of $f$ & $+\infty$ \\
\hline
\end{tabular}
\end{center}
Statement B: The equation $f(x) = -2$ has two solutions in $\mathbb{R}$.
  \item Statement C:
$$\lim_{x \rightarrow +\infty} \frac{\ln(x) - x^{2} + 2}{3x^{2}} = -\frac{1}{3}.$$
  \item Consider the function $k$ defined and continuous on $\mathbb{R}$ by
$$k(x) = 1 + 2\mathrm{e}^{-x^{2}+1}$$
Statement D: There exists a primitive of the function $k$ that is decreasing on $\mathbb{R}$.
  \item Consider the differential equation
$$(E): \quad 3y' + y = 1.$$
Statement E: The function $g$ defined on $\mathbb{R}$ by
$$g(x) = 4\mathrm{e}^{-\frac{1}{3}x} + 1$$
is a solution of the differential equation $(E)$ with $g(0) = 5$.
  \item Statement F: Integration by parts allows us to obtain:
$$\int_{0}^{1} x\mathrm{e}^{-x}\,\mathrm{d}x = 1 - 2\mathrm{e}^{-1}$$
\end{enumerate}