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% Language variables for French exams
\def\professor{R. MOSAID}  % Should be R. MOSAID not {{R. MOSAID}}
\def\classname{2BAC.PC/SVT}
\def\examtitle{Devoir 03 - S01}
\def\schoolname{Lycée : Taghzirt}
\def\academicyear{2025/2026}
\def\subject{Mathématiques}
\def\duration{2h}
\def\secondtitle{\small{visit ~~\wsite~~ for more!}}
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\begin{document}



% Exercise 1 (originally 1)
\printexo{1}{}{
\noindent\textbf{Première partie}:
Soit \( h \) la fonction définie sur \( ]0, +\infty[ \) par :
~\(
h(x) = x - \ln(x)
\)~
\begin{enumerate}[label=\arabic*.]
    \item Calculer \( \lim_{x \to 0^+} h(x) \) et \( \lim_{x \to +\infty} h(x) \)
    \item Calculer \( h'(x) \) pour tout \( x \) de \( ]0, +\infty[ \) puis dresser le tableau de variations de \( h \)
    \item En déduire que \( (\forall x \in ]0, +\infty[) : x - \ln x \geq 1 \)
\end{enumerate}
\noindent\textbf{Deuxième partie}:
Soit \( f \) la fonction numérique définie par :
~\(
\begin{cases}
f(x) = \dfrac{x}{x - \ln x} & \text{si } x > 0 \\
f(0) = 0
\end{cases}
\)~\\
Et soit \( (\mathscr{C}_f) \) sa courbe représentative dans un repère orthonormé \( (O, \vec{i}, \vec{j}) \)
\begin{enumerate}[label=\arabic*.]
    \item Vérifier que \( D_f = [0, +\infty[ \)
    \item Calculer \( \lim_{x \to +\infty} f(x) \) et interpréter le résultat graphiquement
    \item Montrer que \( f \) est continue sur \( [0, +\infty[ \)
    \item Étudier la dérivabilité de \( f \) à droite en 0 et interpréter graphiquement le résultat.
    \item Montrer que \( (\forall x \in ]0, +\infty[) : f'(x) = \dfrac{1 - \ln x}{(x - \ln x)^2} \)
    \item Étudier le signe de \( f'(x) \) puis dresser le tableau de variations de \( f \)
    \item Montrer que la courbe \( (\mathscr{C}_f) \) coupe la droite \( (\Delta) \) d’équation \( y = x \) en deux points dont on déterminera les coordonnées
    \item Vérifier que \( (\forall x \in ]0, +\infty[) : f(x) - x = \dfrac{x(1 - h(x))}{h(x)} \) puis en déduire la position relative de \( (\mathscr{C}_f) \) par rapport à la droite \( (\Delta) \)
    \item Donner l’équation de \( (T) \) la tangente à \( (\mathscr{C}_f) \) au point d’abscisse 1
    \item Construire \( (\mathscr{C}_f) \)
\end{enumerate}
\noindent\textbf{Troisième partie}:
Soit \( (u_n) \) la suite définie par :
~\(
u_0 = \frac{1}{5} \quad \text{et} \quad u_{n+1} = f(u_n) \quad \text{pour tout } n \in \mathbb{N}
\)~
\begin{enumerate}[label=\arabic*.]
    \item Montrer par récurrence que \( (\forall n \in \mathbb{N}), \; 0 \leq u_n \leq 1 \)
    \item Montrer que la suite \( (u_n) \) est décroissante
    \item En déduire que la suite \( (u_n) \) est convergente et déterminer sa limite
\end{enumerate}
}

% Exercise 2 (originally 2)
\printexo{2}{}{
\vspace*{-0.5cm}
\begin{enumerate}[label=\arabic*.]
    \item Résoudre dans \( \mathbb{R} \) : \\
    l’équation : \( (E) : \ln^2 x - 6 \ln x + 5 = 0 \)~
    et l’inéquation : \( (I) : \ln(4x + 5) \leq \ln(3 - x) \)

    \item Déterminer, dans chaque cas, les fonctions primitives de la fonction \( f \) sur l’intervalle \( I \) :
      \begin{itemize}[label=\textbullet]
        \item \( f(x) = 2x - 5 + \dfrac{1}{x^2} \) ;\quad  \( I = \mathbb{R}^*_+ \)
        \quad ; \quad \textbullet~  \( f(x) = \dfrac{x + 4}{x + 2} \) ;\quad  \( I = \left]-2; +\infty\right[ \)
        \item \( f(x) = \dfrac{x + 1}{\sqrt{x^2 + 2x + 7}} \) ; \quad \( I = \mathbb{R} \)
        \quad ;\quad \textbullet~  \( f(x) = \dfrac{\ln x}{x} \) ; \quad  \( I = \mathbb{R}^*_+ \)
    \end{itemize}
\end{enumerate}
}






\end{document}