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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 01 - S01}
\def\schoolname{Lycée : Taghzirt}
\def\academicyear{2025/2026}
\def\subject{Mathématiques}
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\begin{document}



% Exercise 1 (originally 1)
\printexo{1}{}{
\noindent\textbf{\underline{Partie A}}:
Soit \( g \) la fonction définie sur \( ]0; +\infty[ \) par \( g(x) = x \ln(x) - x + 1 \).
\begin{enumerate}[tight]
    \item Calculer \( \lim_{x \to +\infty} g(x) \) et \( \lim_{x \to 0^+} g(x) \).
    \item
    \begin{enumerate}[tight]
        \item Calculer \( g'(x) \) pour tout \( x \) de \( ]0; +\infty[ \) puis dresser le tableau de variation de \( g \).
        \item Déduire le signe de \( g(x) \) sur \( ]0; +\infty[ \).
    \end{enumerate}
\end{enumerate}
\noindent\textbf{\underline{Partie B}}:
Soit \( h \) la fonction définie sur \( ]0; +\infty[ \) par \( h(x) = x \ln^2(x) + 2x \ln(x) + 1 \).
\begin{enumerate}[tight]
    \item Montrer que \( (\forall x \in ]0; +\infty[ ) : h(x) = g(x) + x \left[ \left( \ln(x) + \frac{1}{2} \right)^2 + \frac{3}{4} \right] \).
    \item Déduire que \( (\forall x \in ]0; +\infty[ ) : h(x) > 0 \).
\end{enumerate}
\noindent\textbf{\underline{Partie C}}:
Soit \( f \) la fonction définie sur \( ]0; +\infty[ \) par \( f(x) = x \left( \ln x \right)^2 + \ln(x) + 1 \) et \( (C_f) \) sa courbe représentative dans un repère orthonormé \( (O, \vec{i}, \vec{j}) \).

\begin{enumerate}[tight]
    \item Calculer \( \lim_{x \to +\infty} f(x) \) puis déterminer la branche infinie de \( (C_f) \) au voisinage de \( +\infty \).
    \item
    \begin{enumerate}[tight]
        \item Montrer que \( \lim_{x \to 0^+} x \left( \ln x \right)^2 = 0 \).
        \item Calculer \( \lim_{x \to 0^+} f(x) \) puis interpréter géométriquement le résultat.
    \end{enumerate}
    \item Montrer que pour tout \( x \in ]0; +\infty[ : f'(x) = \frac{h(x)}{x} \).
    \item Donner le tableau de variation de \( f \).
    \item Montrer que l’équation \( f(x) = 0 \) admet une unique solution \( \alpha \) sur \( \left] \frac{1}{e^2}, \frac{1}{e} \right[ \).
    \item Soit \( (\Delta) \) la tangente à \( (C_f) \) au point d’abscisse \( x_0 = 1 \).
    \begin{enumerate}[tight]
        \item Vérifier que l’équation de \( (\Delta) \) est \( (\Delta) : y = x \).
        \item Montrer que \( (\forall x \in ]0; +\infty[ ) : f(x) - x = (\ln(x) + 1)g(x) \).
        \item Étudier la position relative de \( (C_f) \) et de la droite \( (\Delta) : y = x \).
    \end{enumerate}
    \item Construire \( (C_f) \) et \( (\Delta) \) dans le repère \( (O, \vec{i}, \vec{j}) \).
\end{enumerate}
\noindent\textbf{\underline{Partie D}}:
Soit \( (U_n)_{n \in \mathbb{N}} \) la suite numérique définie par :
~$U_0=\frac{1}{2}$~ et ~$U_{n+1}=f(U_n)$~
\begin{enumerate}[tight]
    \item Montrer par récurrence que \( (\forall n \in \mathbb{N}) : \frac{1}{e} \leq U_n \leq 1 \).
    \item Montrer que \( (U_n)_{n \in \mathbb{N}} \) est croissante.
    \item Déduire que \( (U_n)_{n \in \mathbb{N}} \) est convergente puis déterminer sa limite.
\end{enumerate}
}

% Exercise 2 (originally 2)
\printexo{2}{}{
Les questions 1, 2 et 3 sont indépendantes.
\begin{enumerate}[tight]
    \item Déterminer les fonctions primitives des fonctions suivantes :
    \begin{enumerate}[tight]
        \item \( f(x) = x^3 + \frac{x}{\left( x^2 + 1 \right)^2} \) sur \( I = \mathbb{R} \).
        \item \( g(x) = \sqrt{2x + 1} + \frac{1}{\sqrt{3x + 2}} + \frac{\ln^3(x)}{x} \) sur \( I = ]0, +\infty[ \).\\
    \end{enumerate}
    \item
    \begin{enumerate}[tight]
        \item Vérifier que la fonction \( x \mapsto x \ln(x) - x \) est une primitive de la fonction \( x \mapsto \ln(x) \) sur l’intervalle \( ]0, +\infty[ \).
        \item Déterminer la fonction primitive \( H \) de la fonction \( h \) sur \( ]0, +\infty[ \) telle que :
        ~\(
        h(x) = x^3 + \ln(x)
        \)~
        et \( H(1) = 0 \).
    \end{enumerate}
    \item Résoudre dans \( \mathbb{R} \) :
    ~\(
    (I) : \frac{2 + \ln(x)}{\ln(x) - 1} \leq 2 \quad \text{et} \quad (E) : \log^2(x - 1) + 3 \log\left(\frac{1}{x - 1}\right) + 2 = 0.
    \)~
\end{enumerate}
}






\end{document}