\begin{enumerate}
\item
  We consider the functions $x \mapsto \cos x$ and
  $x \mapsto -\sin x$ on $[-\pi; \pi]$ and their
  graphic representations below:

\begin{center}
\includegraphics[width=5.54167in,height=2.57014in]{S7P5/2022_Annals0/2022_04_S7P5_Annals0_Subject2_ExoB3_IMG1.jpg}
\end{center}

Justify that the only solutions of the equation $\cos x + \sin x = 0$
on $[-\pi; \pi]$ are $\dfrac{-\pi}{4}$ and $\dfrac{3\pi}{4}$.

\item
  Let $f$ be the function defined on $[-\pi; \pi]$ by:
  $f(x) = \e^{x} \cdot \sin x$

We note $\mathcal{C}_{f}$ its representative curve in a coordinate system.

\begin{enumerate}
\item
  Determine the variations of the function $f$ on
  $[-\pi; \pi]$, specifying the abscissa, the value and the
  nature of each extremum.

\item
  Determine an equation of the tangent to the curve $\mathcal{C}_{f}$ at the point
  of abscissa $\dfrac{\pi}{2}$.

\item
  On what interval is $\mathcal{C}_{f}$ entirely above
  each of its tangents? To justify.
  
\item
  Using two successive integrations by parts, calculate the
  exact value of the integral:
  
$$\displaystyle\int_{-\pi}^{\pi} f(x) \dx.$$
\end{enumerate}
\end{enumerate}
