\textbf{Part A:}\\
Let $g$ be the function defined on $\mathbb{R}$ by:
$$g(x) = 2\mathrm{e}^{\frac{-1}{3}x} + \frac{2}{3}x - 2$$
\begin{enumerate}
  \item We admit that the function $g$ is differentiable on $\mathbb{R}$ and we denote $g^{\prime}$ its derivative function. Show that, for every real number $x$:
$$g^{\prime}(x) = \frac{-2}{3}e^{-\frac{1}{3}x} + \frac{2}{3}.$$
  \item Deduce the direction of variation of the function $g$ on $\mathbb{R}$.
  \item Determine the sign of $g(x)$, for every real $x$.
\end{enumerate}

\textbf{Part B:}
\begin{enumerate}
  \item Consider the differential equation
$$(E):\quad 3y^{\prime} + y = 0.$$
Solve the differential equation (E).
  \item Determine the particular solution whose representative curve, in a coordinate system of the plane, passes through the point $\mathrm{M}(0;2)$.
  \item Let $f$ be the function defined on $\mathbb{R}$ by:
$$f(x) = 2\mathrm{e}^{-\frac{1}{3}x}$$
and $\mathscr{C}_f$ its representative curve.\\
  a. Show that the tangent line $(\Delta_0)$ to the curve $\mathscr{C}_f$ at the point $\mathrm{M}(0;2)$ has an equation of the form:
$$y = -\frac{2}{3}x + 2$$
  b. Study, on $\mathbb{R}$, the position of this curve $\mathscr{C}_f$ relative to the tangent line $(\Delta_0)$.
\end{enumerate}

\textbf{Part C:}
\begin{enumerate}
  \item Let A be the point on the curve $\mathscr{C}_f$ with abscissa $a$, where $a$ is any real number. Show that the tangent line $(\Delta_a)$ to the curve $\mathscr{C}_f$ at point A intersects the $x$-axis at a point P with abscissa $a+3$.
  \item Explain the construction of the tangent line $(\Delta_{-2})$ to the curve $\mathscr{C}_f$ at point B with abscissa $-2$.
\end{enumerate}