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% --- Basic Settings ---
% Language variables for French exams
\def\professor{R. MOSAID}  % Should be R. MOSAID not {{R. MOSAID}}
\def\classname{2BAC.PC/SVT}
\def\examtitle{Série 3, par Prof: MOHAMED JOUAMRI}
\def\schoolname{\textbf{Lycée :} Lycée : Taghzirt}
\def\academicyear{2025/2026}
\def\subject{Mathématiques}
\def\duration{2h}
\def\secondtitle{\small{(Dérivation et étude des fonctions)}}
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\begin{document}



% Exercise 1
\printexo{1}{}{
\textbf{Partie A}\\
Soit \( g \) la fonction numérique définie sur \( \mathbb{R} \) par :
~$
g(x) = x^3 - 3x - 3
$~

\begin{enumerate}
    \item Étudier les variations de la fonction \( g \) sur \( \mathbb{R} \).
    \item
    \begin{enumerate}
        \item Montrer que l’équation \( g(x) = 0 \) admet une solution unique \( \alpha \) dans \([1, +\infty[\), et que :
        ~$
        2 < \alpha < 3
        $~
        \item En déduire le signe de \( g(x) \) sur \( \mathbb{R} \).
    \end{enumerate}
\end{enumerate}

\textbf{Partie B}\\
Soit \( f \) la fonction numérique définie sur \( \mathbb{R}^* \) par :
~$
f(x) = x + \frac{3}{x} + \frac{3}{2x^2}
$~

\begin{enumerate}
    \item Calculer les limites de la fonction \( f \) aux bornes de \( D_f \) et
    ~$
    \lim_{x \to \infty} \left( f(x) - x \right)
    $~
    \item Déterminer les branches infinies de \( (\mathscr C_f) \).
    \item
    \begin{enumerate}
        \item Montrer que :
        ~$
        \forall x \in \mathbb{R}^*, \quad f'(x) = \frac{g(x)}{x^3}
        $~
        \item Dresser le tableau de variations de la fonction \( f \).
    \end{enumerate}
    \item Étudier la position relative de \( (\mathscr C_f) \) et la droite \( (\Lambda) \) d’équation \( y = x \).
    \item Étudier la concavité de \( (\mathscr C_f) \).
    \item Tracer \( (\mathscr C_f) \) dans un repère orthonormé \( (O, \vec{i}, \vec{j}) \). (On donne \( \alpha = 2.1 \) et \( f(\alpha) = 3.8 \))
\end{enumerate}
}

% Exercise 2
\printexo{2}{}{
On considère la fonction numérique \( f \) définie sur \( \mathbb{R}^+ \) par :
~$
f(x) = x \left( \sqrt{x} - 2 \right)^2
$~\\
On désigne par \( (\mathscr C_f) \) la courbe représentative de la fonction \( f \) dans un repère orthonormé \( (O, \vec{i}, \vec{j}) \).
\begin{enumerate}
    \item Calculer \( \displaystyle \lim_{x \to +\infty} f(x) \) puis \( \displaystyle \lim_{x \to +\infty} \frac{f(x)}{x} \). Que peut-on déduire ?
    \item Étudier la dérivabilité de \( f \) à droite en 0 puis interpréter graphiquement le résultat obtenu.
    \item Montrer que :
    ~$
    \forall x \in \mathbb{R}_+^*, \quad f'(x) = 2(\sqrt{x} - 1)(\sqrt{x} - 2)
    $~\\
    puis étudier les variations de la fonction \( f \) sur \( \mathbb{R}_+ \).
    \item Étudier la concavité de la courbe \( (\mathscr C_f) \) et montrer que \( (\mathscr C_f) \) admet un unique point d’inflexion \( I \) auquel on déterminera ses coordonnées.
    \item Résoudre dans \( \mathbb{R}^+ \) l’équation \( f(x) = x \) et interpréter le résultat graphiquement.
    \item Tracer la courbe \( (\mathscr C_f) \) dans le repère \( (O, \vec{i}, \vec{j}) \).
    \item Soit \( g \) la restriction de la fonction \( f \) sur l’intervalle \( I = [4, +\infty[ \).
    \begin{enumerate}
        \item Montrer que \( g \) admet une fonction réciproque \( g^{-1} \) définie sur un intervalle \( J \) à déterminer.
        \item Dresser le tableau de variations de la fonction \( g^{-1} \).
        \item Calculer \( g(9) \) puis déterminer \( (g^{-1})'(9) \).
        \item Trouver \( g^{-1}(x) \) pour tout \( x \in J \).
    \end{enumerate}
    \item Tracer la courbe \( (\mathscr C_{g^{-1}}) \) dans le repère \( (O, \vec{i}, \vec{j}) \).
\end{enumerate}
}






% Bottom message
\begin{center}
\normalsize{ \vskip 1pt \hrule height 1pt \vskip 2pt \RL{\arabicfont ﴿وَإِنَّ رَبَّكَ لَذُو فَضْلٍ عَلَى ٱلنَّاسِ وَلَـٰكِنَّ أَكْثَرَهُمْ لَا يَشْكُرُونَ﴾ (النمل 73)} }
\end{center}


\end{document}