An advertiser wishes to print a logo on a T-shirt. He draws this logo using the curves of two functions $f$ and $g$ defined on $\mathbb{R}$ by:
$$f(x) = \mathrm{e}^{-x}(-\cos x + \sin x + 1) \text{ and } g(x) = -\mathrm{e}^{-x}\cos x$$
It is admitted that the functions $f$ and $g$ are differentiable on $\mathbb{R}$.

\section*{Part A — Study of function $f$}
\begin{enumerate}
  \item Justify that, for all $x \in \mathbb{R}$:
  $$-\mathrm{e}^{-x} \leqslant f(x) \leqslant 3\mathrm{e}^{-x}$$
  \item Deduce the limit of $f$ as $x \to +\infty$.
  \item Prove that, for all $x \in \mathbb{R}$, $f'(x) = \mathrm{e}^{-x}(2\cos x - 1)$ where $f'$ is the derivative of $f$.
  \item In this question, we study function $f$ on the interval $[-\pi; \pi]$.\\
  a. Determine the sign of $f'(x)$ for $x$ in the interval $[-\pi; \pi]$.\\
  b. Deduce the variations of $f$ on $[-\pi; \pi]$.
\end{enumerate}

\section*{Part B — Area of the logo}
We denote by $\mathscr{C}_f$ and $\mathscr{C}_g$ the graphs of functions $f$ and $g$ in an orthonormal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath})$. The graphical unit is 2 centimetres.

\begin{enumerate}
  \item Study the relative position of curve $\mathscr{C}_f$ with respect to curve $\mathscr{C}_g$ on $\mathbb{R}$.
  \item Let $H$ be the function defined on $\mathbb{R}$ by:
  $$H(x) = \left(-\frac{\cos x}{2} - \frac{\sin x}{2} - 1\right)\mathrm{e}^{-x}$$
  It is admitted that $H$ is an antiderivative of the function $x \mapsto (\sin x + 1)\mathrm{e}^{-x}$ on $\mathbb{R}$.\\
  We denote by $\mathscr{D}$ the region bounded by curve $\mathscr{C}_f$, curve $\mathscr{C}_g$ and the lines with equations $x = -\frac{\pi}{2}$ and $x = \frac{3\pi}{2}$.\\
  a. Shade the region $\mathscr{D}$ on the graph in the appendix to be returned with your work.\\
  b. Calculate, in square units, the area of region $\mathscr{D}$, then give an approximate value to $10^{-2}$ in $\mathrm{cm}^2$.
\end{enumerate}