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% --- Basic Settings ---
\def\professor{R. MOSAID}
\def\classname{2BAC.SM}
\def\examtitle{Devoir {\small par SAMIR LAKHRISSI}}
\def\schoolname{\textbf{Lycée :} Taghzirt}
\def\academicyear{2025/2026}
\def\subject{Mathématiques}
\def\duration{2h}
\def\secondtitle{\small(Dérivation et étude des fonctions)}
\def\province{Direction provinciale de\\ Beni Mellal}
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\begin{document}
\vspace*{-1cm}
% Exercise 1
\printexo{1}{}{

\vspace*{-0.5cm}
\noindent On considère la fonction :
~$
\begin{cases}
  f(x) = \pi \frac{x - \arctan x}{\left| \arctan x \right|} ; \quad x \neq 0\\
  f(0) = 0
\end{cases}
$~\\
\textbf{Partie I :}
\begin{enumerate}[itemsep=0pt, topsep=0pt, parsep=0pt, partopsep=0pt]
    \item Vérifier que \( f \) est impaire
    \item Étudier la continuité de \( f \) en 0
    \item
    \begin{enumerate}
        \item Calculer la dérivée de la fonction : \( g(x) = \arctan(x) + \arctan\left(\frac{1}{x}\right) \) sur \( ]0; +\infty[\)
        \item En déduire que : \( (\forall x > 0),\quad \arctan x + \arctan \frac{1}{x} = \frac{\pi}{2} \)
        \item Montrer que la droite \( (\Delta) : y = 2x + \frac{4}{\pi} - \pi \) est une asymptote oblique de \( (C_f) \) au voisinage de \( +\infty \)
    \end{enumerate}
\end{enumerate}
\textbf{Partie II :}
\begin{enumerate}[itemsep=0pt, topsep=0pt, parsep=0pt, partopsep=0pt]
    \item
    \begin{enumerate}
        \item En utilisant le théorème des accroissements finis, montrer que :\\
        ~$(\forall x > 0), \quad 0 < \frac{x - \arctan x}{x} < \frac{x^2}{1 + x^2}$~
        \item En déduire que : \( (\forall x > 0), \quad \arctan x > \frac{x}{1 + x^2} \)
    \end{enumerate}
    \item Étudier la dérivabilité de \( f \) en 0 à droite, puis interpréter le résultat graphiquement
    \item Calculer \( f'(x) \) pour tous \( x \in ]0; +\infty[ \), puis tracer le tableau de variation de \( f \)
    \item Tracer \( (C_f) \)
\end{enumerate}
}

% Exercise 2
\printexo{2}{}{

\noindent On considère la fonction : \( f(x) = \frac{x^2 + x + 1}{x^2 + 1} - \arctan x \)\\
\textbf{Partie I :}
\begin{enumerate}[itemsep=0pt, topsep=0pt, parsep=0pt, partopsep=0pt]
    \item Calculer \( \lim_{x \to +\infty} f(x) \) et \( \lim_{x \to -\infty} f(x) \)
    \item Calculer \( f'(x) \) pour tous \( x \in \mathbb{R} \), puis tracer le tableau de variation de \( f \)
    \item Montrer que : \( (\forall x \in \mathbb{R}),\quad |f'(x)| \leq \frac{1}{2} \)
    \item Vérifier que : \( (\forall x \in \mathbb{R}), \quad f''(x) = \frac{4x(x^2 - 1)}{(x^2 + 1)^3} \),~ puis déterminer les points d'inflexion
    \item Tracer \( (C_f) \)
\end{enumerate}
\textbf{Partie II :}
On considère la suite définie par :
~$
\begin{cases}
u_0 \in \mathbb{R} \\
u_{n+1} = f(u_n); \, n \geq 0
\end{cases}
$~
\begin{enumerate}[itemsep=0pt, topsep=0pt, parsep=0pt, partopsep=0pt]
    \item Montrer que : \( (\exists ! \alpha \in \mathbb{R})\quad f(\alpha) = \alpha \)
    \item Montrer que : \( (\forall n \in \mathbb{N}),\quad |u_{n+1} - \alpha| \leq \frac{1}{2} |u_n - \alpha| \), puis déterminer \( \lim_{n \to +\infty} u_n \)
\end{enumerate}
}
\vspace*{-0.5cm}
\begin{center}
  \vskip 3pt \hrule height 3pt \vskip 5pt \RL{\arabicfont ﴿وَكَذَٰلِكَ أَنزَلْنَـٰهُ ءَايَـٰتٍۭ بَيِّنَـٰتٍ وَأَنَّ ٱللَّهَ يَهْدِى مَن يُرِيدُ (16)•  إِنَّ ٱلَّذِينَ ءَامَنُوا۟ وَٱلَّذِينَ هَادُوا۟ وَٱلصَّـٰبِـِٔينَ وَٱلنَّصَـٰرَىٰ وَٱلْمَجُوسَ وَٱلَّذِينَ أَشْرَكُوٓا۟ إِنَّ ٱللَّهَ يَفْصِلُ بَيْنَهُمْ يَوْمَ ٱلْقِيَـٰمَةِ ۚ إِنَّ ٱللَّهَ عَلَىٰ كُلِّ شَىْءٍ شَهِيدٌ (17)•  أَلَمْ تَرَ أَنَّ ٱللَّهَ يَسْجُدُ لَهُۥ مَن فِى ٱلسَّمَـٰوَٰتِ وَمَن فِى ٱلْأَرْضِ وَٱلشَّمْسُ وَٱلْقَمَرُ وَٱلنُّجُومُ وَٱلْجِبَالُ وَٱلشَّجَرُ وَٱلدَّوَآبُّ وَكَثِيرٌ مِّنَ ٱلنَّاسِ ۖ وَكَثِيرٌ حَقَّ عَلَيْهِ ٱلْعَذَابُ ۗ وَمَن يُهِنِ ٱللَّهُ فَمَا لَهُۥ مِن مُّكْرِمٍ ۚ إِنَّ ٱللَّهَ يَفْعَلُ مَا يَشَآءُ ۩ (18)• ﴾ (الحج الآيات 16-18) }
\end{center}


\end{document}