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% --- Basic Settings ---
% Language variables for French exams
\def\professor{R. MOSAID}  % Should be R. MOSAID not {{R. MOSAID}}
\def\classname{1BAC.SM}
\def\examtitle{Devoir 03 - S01, Par: MR. EL ANANI}
\def\schoolname{Lycée : Taghzirt}
\def\academicyear{2025/2026}
\def\subject{Mathématiques}
\def\duration{2h}
\def\secondtitle{\small{visit ~~\wsite~~ for more!}}
\def\province{Direction provinciale\newline de Beni Mellal}
\def\logo{\includegraphics[width=\linewidth]{images/logo-men.png}}
\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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% Custom exercise environments (uncomment if needed)
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% Additional packages (uncomment if needed)
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% ----------------------------------------------------------------------
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\begin{document}



% Section for Exercise 1
\noindent
\hbox to \textwidth{
  \hfil
  \fcolorbox{red}{yellow!25}{%
    \begin{minipage}{0.9\textwidth}
      \centering
      Conseils clés : Présentation soignée + Vérification systématique
    \end{minipage}%
  }%
  \hfil
}%
\vspace{0.8em}

% Exercise 1 (originally 1)
\printexo{1}{% : ~ (8 points)
 : (fonction rationnelle, fonction composée, fonction périodique) 20 points}{
\vspace*{-1cm}
\subsection*{Partie I : Étude d’une fonction rationnelle (8pts)}
Soit \( f \) \textbf{la fonction numérique} définie sur \(\mathbb{R}\) par :
~\(
f(x) = \frac{4x}{x^2 + 1}
\)~
\begin{enumerate}[label=\arabic*)]
    \item Étudier \textbf{la parité} de la fonction \( f \) \dotfill [1pt]
    \item Montrer que pour tout \( x \in \mathbb{R} \) : \( |f(x)| \leq 2 \) \dotfill [1.5pt]
    \item
    \begin{enumerate}[label=\alph*)]
        \item Étudier \textbf{la monotonie} de la fonction \( f \) sur \([0;1]\) et sur \([1;+\infty[\) \dotfill [1pt]
        \item Dresser le tableau de variations complet de la fonction \( f \) sur \(\mathbb{R}\) \dotfill [1pt]
    \end{enumerate}
    \item Soient \( a \) et \( b \) deux réels positifs. Montrer que :
    ~\(
    a + b \geq \sqrt{2} \Rightarrow \frac{(a + b)^2 + 1}{a + b} \geq \frac{3\sqrt{2}}{2} \dotfill [1.5pt]
    \)~
    \item Montrer que : \( f(\mathbb{R}) = [-2;2] \) \dotfill [2pts]
\end{enumerate}

\subsection*{Partie II : Composition et fonction racine carrée (6pts)}
Soit \( g \) et \( h \) \textbf{les fonctions numériques} définies sur \(\mathbb{R}^+\) par :
~\(
g(x) = \sqrt{x} \quad \text{et} \quad h(x) = \frac{4\sqrt{x}}{x+1}.
\)~

\begin{enumerate}[label=\arabic*)]
  \item \textbf{Représenter} \( g \) dans un repère orthonormé $(O; \vec i, \vec j)$ \dotfill [1pt]
    \item
    \begin{enumerate}[label=\alph*)]
        \item Montrer que : \( (\forall x \in \mathbb{R}^+)\quad  ; 0 \leq h(x) \leq 2 \) \dotfill [1pt]
        \item Résoudre dans \(\mathbb{R}^+\) l’équation : \( h(x) = g(x) \) \dotfill [1pt]
        \item Vérifier que : \( (\forall x \in \mathbb{R}^+) \quad  ; h(x) = f \circ g(x) \) \dotfill [1pt]
        \item Dresser le tableau de variations de la fonction \( h \) \dotfill [2pts]
    \end{enumerate}
\end{enumerate}

\subsection*{Partie III : Fonction périodique avec partie entière (6pts)}

On considère maintenant \textbf{la fonction} \( \varphi \) définie par :
~\(
\varphi(x) = 1 + \sqrt{x - 4E\left( \frac{x}{4} \right)}
\)~

où \( E \) désigne \textbf{la fonction partie entière} et \((C_\varphi)\) sa courbe représentative dans $(O; \vec i, \vec j)$

\begin{enumerate}[label=\arabic*)]
    \item Montrer que : \( D_\varphi = \mathbb{R} \) \dotfill [1pt]
    \item Montrer que le nombre 4 est une période de la fonction \( \varphi \) \dotfill [1pt]
    \item Montrer que : \( (\forall x \in \mathbb{R}) \quad  ; 1 \leq \varphi(x) < 3 \) \dotfill [1pt]
    \item
    \begin{enumerate}[label=\alph*)]
        \item Vérifier que pour tout \( x \in [0;4[; \varphi(x) = 1 + \sqrt{x} \) \dotfill [1pt]
        \item \textbf{Tracer} la courbe \((C_\varphi)\) sur l’intervalle \([-4;8[\) \dotfill [2pts]
    \end{enumerate}
\end{enumerate}
}





% Bottom message
\begin{center}
  \normalsize{%
    \vskip 3pt \hrule height 3pt \vskip 5pt
    \RL{\arabicfont
    ﴿وَأَقْسَمُوا۟ بِٱللَّهِ جَهْدَ أَيْمَـٰنِهِمْ ۙ لَا يَبْعَثُ ٱللَّهُ مَن يَمُوتُ ۚ بَلَىٰ وَعْدًا عَلَيْهِ حَقًّا وَلَـٰكِنَّ أَكْثَرَ ٱلنَّاسِ لَا يَعْلَمُونَ (38)•  لِيُبَيِّنَ لَهُمُ ٱلَّذِى يَخْتَلِفُونَ فِيهِ وَلِيَعْلَمَ ٱلَّذِينَ كَفَرُوٓا۟ أَنَّهُمْ كَانُوا۟ كَـٰذِبِينَ (39)• ﴾ (النحل الآيات 38-39)
    }%
  }%
\end{center}

\end{document}