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% --- Basic Settings ---
\def\professor{R. MOSAID}
\def\classname{1BAC.SM}
\def\examtitle{Devoir 02 - S01}
\def\schoolname{\textbf{Lycée :} Taghzirt}
\def\academicyear{2025/2026}
\def\subject{Mathématiques}
\def\duration{2h}
\def\secondtitle{\small(Généralités sur les fonctions \& Ensembles)}
\def\province{Direction provinciale de\\ Beni Mellal}
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\def\wsite{\href{https://www.mosaid.xyz}{\textcolor{magenta}{\texttt{www.mosaid.xyz}}}}
\def\ddate{\hfill \number\day/\number\month/\number\year~~}
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\begin{document}

% Exercise 1
\printexo{1}{}{
On considère les fonctions: \( f(x) = x^2 + 2x + 2 \) et \( g(x) = \frac{4x + 2}{x - 1} \)
\begin{enumerate}
    \item
    \begin{enumerate}[label=(\alph*)]
        \item Donner les tableaux de variations de \( f \) et de \( g \).\dotfill \textbf{(1pt)}
        \item Quelle est la nature de \( \mathcal C_f \) et \( \mathcal C_g \)?\dotfill \textbf{(1pt)}
    \end{enumerate}

    \item
    \begin{enumerate}[label=(\alph*)]
        \item Développer \((x + 1)(x^2 - 4)\) puis déduire les points d’intersections de \( \mathcal C_f \) et \( \mathcal C_g \).\dotfill \textbf{(1,5pt)}
        \item Tracer dans un même repère orthonormé \( \mathcal C_f \) et \( \mathcal C_g \).\dotfill \textbf{(1,5pt)}
        \item Résoudre graphiquement \( \frac{x + 2}{x - 1} \geq \frac{x^2}{2} + x \).\dotfill \textbf{(1pt)}
    \end{enumerate}

    \item Soit \( F \) la fonction définie par: \( F(x) = x + 2\sqrt{x - 2} \).
    \begin{enumerate}[label=(\alph*)]
        \item Déterminer la fonction \( h \) telle que: \( F(x) = f \circ h(x) \).\dotfill \textbf{(0,75pt)}
        \item Étudier le sens de variation de \( F \) sur \([2,+\infty[\).\dotfill \textbf{(0,75pt)}
    \end{enumerate}
\end{enumerate}
}

% Exercise 2
\printexo{2}{}{
Soit \( m \) un paramètre de \( \mathbb{R}_+^* \). On considère la fonction \( f_m \) définie sur \( D = \mathbb{R}_+^* \) par:
~$ f_m(x) = \frac{2x}{m^2} + \frac{m}{x^2} $~
\begin{enumerate}
    \item
    \begin{enumerate}[label=(\alph*)]
        \item Montrer que \( (\forall (x,y) \in D^2) \quad f_m(x) - f_m(y) = (x - y)\left( \frac{2}{m^2} - \frac{m(x + y)}{x^2y^2} \right) \).\dotfill \textbf{(1pt)}
        \item Montrer que \( f_m \) est croissante sur \([m,+\infty[\) et décroissante sur \(]0,m]\).\dotfill \textbf{(1,5pt)}
    \end{enumerate}

    \item En déduire que: \( (\forall x \in D) \quad f_m(x) \geq \frac{3}{m} \).\dotfill \textbf{(1pt)}

    \item Déduire de ce qui précède que:\dotfill \textbf{(1,5pt)}\\
    ~$ (\forall (a,b,c) \in D^3) \quad \frac{a + 2c}{b^2} + \frac{a + 2c}{b^2} + \frac{a + 2c}{b^2} \geq 3\left( \frac{1}{a} + \frac{1}{b} + \frac{1}{c} \right) $~\\
    \textcolor{red}{peut etre cette question doit etre: }~\\
    ~$ (\forall (a,b,c) \in D^3) \quad \frac{a + 2c}{b^2} + \frac{b + 2a}{c^2} + \frac{c + 2b}{a^2} \geq 3\left( \frac{1}{a} + \frac{1}{b} + \frac{1}{c} \right) $~\\
    \textcolor{red}{à vérifier.}~
\end{enumerate}
}

% Exercise 3
\printexo{3}{}{
\begin{enumerate}
    \item Soit \( \alpha \in \mathbb{R} \). On considère l’ensemble suivant: \( I_\alpha = \{x \in \mathbb{R} / |3x - \alpha| < 2\} \).
    \begin{enumerate}[label=(\alph*)]
        \item Déterminer l’ensemble \( I_\alpha \).\dotfill \textbf{(1pt)}
        \item Déterminer les valeurs de \( \alpha \) telles que: \( I_\alpha \cap [0,1] = \emptyset \).\dotfill \textbf{(1pt)}
        \item Déterminer les valeurs de \( \alpha \) telles que: \( I_\alpha \subset [1,2] \).\dotfill \textbf{(1pt)}
    \end{enumerate}

    \item On considère les deux ensembles suivants:\\
    ~$ A = \left\{(n,m) \in \mathbb{Z}^* \times \mathbb{Z}^* ~/~ \frac{1}{n} + \frac{1}{m} = \frac{1}{5}\right\} $~\quad \quad ;\quad \quad
    ~$ B = \{x^2 - 2x + ~/~ x \in [-1,2]\} $~
    \begin{enumerate}[label=(\alph*)]
        \item Montrer que \( A \neq \emptyset \) et \( B \neq \emptyset \).\dotfill \textbf{(0,5pt)}
        \item A-t-on \( 0 \in B \) ? Qu’en est-il de 2?\dotfill \textbf{(1pt)}
        \item Montrer que \( B \subset [1,5] \). Est-ce que \([1,5] \subset B\) ?\dotfill \textbf{(1pt)}
        \item Montrer que \( (n,m) \in A \iff (n - 5)(m - 5) = 25 \). Puis déterminer \( A \) en extension.\dotfill \textbf{(1pt)}
    \end{enumerate}
\end{enumerate}

\vspace{1cm}
\noindent
\textbf{N.B:} 1 point sera attribué à la présentation de la copie et la bonne rédaction des réponses.
}

\end{document}