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\def\professor{R. MOSAID}  % Should be R. MOSAID not {{R. MOSAID}}
\def\classname{2BAC.PC/SVT}
\def\examtitle{Devoir libre N\s{$\circ$}2 -- Par: Pr. MOHAMED JOUAHRI}
\def\schoolname{\textbf{Lycée :} Lycée : Taghzirt}
\def\academicyear{2025/2026}
\def\subject{Mathématiques}
\def\duration{2h}
\def\secondtitle{\small{(Fonctions primitives,Suites \& Etude des fonctions)}}
\def\province{Direction provinciale\newline de Beni Mellal}
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\def\bottommsg{\\vspace*{-0.5cm}\n\\begin{center}\n  \\RL{\\arabicfont\n  ﴿بِسْمِ ٱللَّهِ ٱلرَّحْمَـٰنِ ٱلرَّحِيمِ طه (1)•  مَآ أَنزَلْنَا عَلَيْكَ ٱلْقُرْءَانَ لِتَشْقَىٰٓ (2)•  إِلَّا تَذْكِرَةً لِّمَن يَخْشَىٰ (3)•  تَنزِيلًا مِّمَّنْ خَلَقَ ٱلْأَرْضَ وَٱلسَّمَـٰوَٰتِ ٱلْعُلَى (4)•  ٱلرَّحْمَـٰنُ عَلَى ٱلْعَرْشِ ٱسْتَوَىٰ (5)•  لَهُۥ مَا فِى ٱلسَّمَـٰوَٰتِ وَمَا فِى ٱلْأَرْضِ وَمَا بَيْنَهُمَا وَمَا تَحْتَ ٱلثَّرَىٰ (6)•  وَإِن تَجْهَرْ بِٱلْقَوْلِ فَإِنَّهُۥ يَعْلَمُ ٱلسِّرَّ وَأَخْفَى (7)•  ٱللَّهُ لَآ إِلَـٰهَ إِلَّا هُوَ ۖ لَهُ ٱلْأَسْمَآءُ ٱلْحُسْنَىٰ (8)• ﴾ (طه الآيات 1-8)\n  }\n\\end{center}\n}
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\begin{document}



% Exercise 1 (originally 1)
\printexo{1}{}{
%\vspace*{-0.75cm}
\begin{enumerate}[label=\arabic*., tight]
    \item Dans chacun des cas suivants, déterminer les primitives de la fonction \( f \) sur l’intervalle \( I = \mathbb{R} \):\\
    ~$
    f(x) = x^2 + 3x\sqrt{x^2 + 2} ; \quad f(x) = \frac{x}{\sqrt{x^2 + 10}} ; \quad f(x) = 3x^2(x^3 + 1)^5 + \sin(2x)
    $~
    \item Soit \( g \) la fonction numérique définie sur l’intervalle \( I = ]1, +\infty[ \) par :
    ~$
    g(x) = \frac{2x + 5}{(x - 1)^3}.
    $~
    \begin{enumerate}[label=\alph*)]
        \item Déterminer les réels \( a \) et \( b \) tels que :
        ~$
        (\forall x \in I) \quad g(x) = \frac{a}{(x - 1)^2} + \frac{b}{(x - 1)^3}.
        $~
        \item En déduire la primitive de la fonction \( g \) sur l’intervalle \( I \) qui s’annule en 2.
    \end{enumerate}
\end{enumerate}
}

% Exercise 2 (originally 2)
\printexo{2}{}{
Soit \((U_n)\) la suite numérique définie par :
~$
U_0 = 0 \quad \text{et} \quad U_{n+1} = \frac{2U_n + 1}{U_n + 2} ; \quad (\forall n \in \mathbb{N})
$~

\begin{enumerate}[label=\arabic*., tight]
    \item Montrer que :
    ~$
    (\forall n \in \mathbb{N}) : \quad 0 \leq U_n < 1.
    $~
    \item Montrer que la suite \((U_n)\) est croissante, en déduire qu’elle est convergente.
    \item Montrer que :
    ~$
    (\forall n \in \mathbb{N}) \quad 1 - U_{n+1} \leq \frac{1}{2}(1 - U_n).
    $~
    \item En déduire que :
    ~$
    (\forall n \in \mathbb{N}) \quad 1 - U_n \leq \left(\frac{1}{2}\right)^n, \text{ puis calculer } \lim U_n.
    $~
    \item Soit \( n \in \mathbb{N} \) on pose :
    ~$
    V_n = \frac{U_n - 1}{U_n + 1}.
    $~
    \begin{enumerate}[label=\alph*)]
        \item Montrer que \((V_n)\) est géométrique de raison \( q = \frac{1}{3} \).
        \item Déterminer \( V_n \) en fonction de \( n \) puis \( U_n \) en fonction de \( n \).
    \end{enumerate}
\end{enumerate}
}

% Exercise 3 (originally 3)
\printexo{3}{}{
\textbf{Partie A}\\
Soit \( f \) la fonction numérique définie sur \( \mathbb{R}_+ \) par :
~$
f(x) = 4x\sqrt{x} - 3x^2.
$~

\begin{enumerate}[label=\arabic*., tight]
    \item Étudier la dérivabilité de \( f \) à droite en 0 puis interpréter graphiquement le résultat obtenu.
    \item  {\fontsize{11}{13}\selectfont Calculer \( \lim_{x \to +\infty} f(x) \) puis \( \lim_{x \to +\infty} \frac{f(x)}{x} \)
      et déduire la nature de la branche infinie de
      \((\mathscr C_f)\)  au voisinage de \( +\infty \).}
    \item  Montrer que :
    ~$
    (\forall x \in \mathbb{R}_+^*) \quad f'(x) = 6\sqrt{x}\left(1 - \sqrt{x}\right),
    $~
     {\fontsize{11}{13}\selectfont puis étudier les variations de la fonction} \( f \) sur \( \mathbb{R}_+ \).
    \item
    \begin{enumerate}[label=\alph*)]
        \item Vérifier que :
        ~$
        (\forall x \in \mathbb{R}_+^*) \quad f(x) - x = x\left(\sqrt{x} - 1\right)\left(1 - 3\sqrt{x}\right).
        $~
        \item Étudier la position relative de \((\mathscr C_f)\) et la droite \((D)\) d’équation \( y = x \).
    \end{enumerate}
    \item Montrer que :
    ~$
    (\forall x \in \mathbb{R}_+^*) \quad f''(x) = \frac{3\left(1 - 2\sqrt{x}\right)}{\sqrt{x}},
    $~
    puis étudier la concavité de \((\mathscr C_f)\).
    \item Déterminer les points d’intersections de \((\mathscr C_f)\) et l’axe des abscisses, puis construire \((\mathscr C_f)\).
\end{enumerate}
\textbf{Partie B}\\
Soit \((U_n)\) la suite numérique définie par :
~$
U_0 = \frac{4}{9} \quad \text{et} \quad (\forall n \in \mathbb{N}) \quad U_{n+1} = f(U_n).
$~

\begin{enumerate}[label=\arabic*., tight]
    \item Montrer que :
    ~$
    (\forall n \in \mathbb{N}) : \quad \frac{1}{9} \leq U_n \leq 1.
    $~
    \item Montrer que la suite \((U_n)\) est croissante puis déterminer \( \lim U_n \).
\end{enumerate}
}






% Bottom message
\vspace*{-0.5cm}
\begin{center}
  \RL{\arabicfont
  ﴿بِسْمِ ٱللَّهِ ٱلرَّحْمَـٰنِ ٱلرَّحِيمِ طه (1)•  مَآ أَنزَلْنَا عَلَيْكَ ٱلْقُرْءَانَ لِتَشْقَىٰٓ (2)•  إِلَّا تَذْكِرَةً لِّمَن يَخْشَىٰ (3)•  تَنزِيلًا مِّمَّنْ خَلَقَ ٱلْأَرْضَ وَٱلسَّمَـٰوَٰتِ ٱلْعُلَى (4)•  ٱلرَّحْمَـٰنُ عَلَى ٱلْعَرْشِ ٱسْتَوَىٰ (5)•  لَهُۥ مَا فِى ٱلسَّمَـٰوَٰتِ وَمَا فِى ٱلْأَرْضِ وَمَا بَيْنَهُمَا وَمَا تَحْتَ ٱلثَّرَىٰ (6)•  وَإِن تَجْهَرْ بِٱلْقَوْلِ فَإِنَّهُۥ يَعْلَمُ ٱلسِّرَّ وَأَخْفَى (7)•  ٱللَّهُ لَآ إِلَـٰهَ إِلَّا هُوَ ۖ لَهُ ٱلْأَسْمَآءُ ٱلْحُسْنَىٰ (8)• ﴾ (طه الآيات 1-8)
  }
\end{center}


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