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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{Devoir 03 - S01, Par Prof Saissi Rachid}
\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 theorem environments (uncomment if needed)
% \newtheorem{theorem}{Theorem}
% \newtheorem{lemma}{Lemma}
% \newtheorem{corollary}{Corollary}

% Custom exercise environments (uncomment if needed)
% \newenvironment{solution}{\begin{proof}[Solution]}{\end{proof}}
% \newenvironment{remark}{\begin{proof}[Remark]}{\end{proof}}

% Custom commands for this exam (add your own below)
% \newcommand{\answerline}[1]{\underline{\hspace{\#1}}}
% \newcommand{\points}[1]{\hfill(\mbox{\#1 points})}

% Additional packages (uncomment if needed)
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% \usepackage{chemformula} % For chemical formulas
% \usepackage{circuitikz} % For circuit diagrams

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% --- Header Style 9 (fr) ---
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\begin{document}



\vspace*{-0.75cm}
% Exercise 1 (originally 1)
\printexo{1}{}{
\begin{enumerate}
    \item Vérifier que : \((x-1)(3-x)=-x^2+4x-3\)
    \item
    \begin{enumerate}[label=\alph*)]
        \item En déduire les solutions dans \(\mathbb{R}\) de l’équation : \((\ln x)^2-4\ln x+3=0\)
        \item Résoudre dans \(\mathbb{R}\) l’inéquation : \((\ln x)^2+3\leq 4\ln x\)
    \end{enumerate}
    \item Calculer :
    ~\(A = \ln \left( \sqrt{2026} - \sqrt{2025} \right)^{2026} + \ln \left( \sqrt{2026} + \sqrt{2025} \right)^{2026}\)~
\end{enumerate}
}

% Exercise 2 (originally 2)
\printexo{2}{}{
\noindent\textbf{I.}
Soit \(h\) la fonction définie sur \(]0; +\infty [\) par :
~\(h(x) = x\ln x - x + 1\)~
\begin{enumerate}
    \item
    \begin{enumerate}[label=\alph*)]
        \item Montrer que \(h'(x) = \ln x\) pour tout \(x\) de \(]0; +\infty [\)
        \item Montrer que \(h\) est décroissante sur \(]0; 1]\) et croissante sur \([1; +\infty [\)
    \end{enumerate}

    \item Calculer \(h(1)\) et en déduire que \(h(x) \geq 0\) pour tout \(x\) de \(]0; +\infty [\)
\end{enumerate}
\noindent\textbf{II.}
Soit \(g\) la fonction définie sur \(]0; +\infty [\) par :
~\(g(x) = x(\ln x)^2 + 2x\ln x + 1\)~
\begin{enumerate}
    \item Montrer que :
    ~\(g(x) = h(x) + x \left[ \left( \ln x + \frac{1}{2} \right)^2 + \frac{3}{4} \right]\)~
    pour tout \(x\) de \(]0; +\infty [\)
    \item En déduire que : \(g(x) > 0\) pour tout \(x\) de \(]0; +\infty [\)
\end{enumerate}
\noindent\textbf{III.}
Soit \(f\) la fonction définie sur \(]0; +\infty [\) par :
~\(f(x) = x(\ln x)^2 + \ln x + 1\)~
\begin{enumerate}
    \item Calculer \(\lim\limits_{x \to +\infty} f(x)\) puis déduire la nature de la branche infinie de \((C)\) au voisinage de \(+\infty\)
    \item
    \begin{enumerate}[label=\alph*)]
        \item Vérifier que : \(x(\ln x)^2 = 4 \left( \sqrt{x} \ln \sqrt{x} \right)^2\) pour tout \(x\) de \(]0; +\infty [\) \\et en déduire que \(\lim\limits_{x \to 0^+} x(\ln x)^2 = 0\)
        \item Calculer \(\lim\limits_{x \to 0^+} f(x)\) puis interpréter géométriquement le résultat.
    \end{enumerate}
    \item
    \begin{enumerate}[label=\alph*)]
        \item Montrer que : \(f'(x) = \frac{g(x)}{x}\) pour tout \(x\) de \(]0; +\infty [\)
        \item En déduire que \(f\) est strictement croissante sur \(]0; +\infty [\)
        \item Montrer qu’il existe un réel unique \(\alpha\) de l’intervalle \(\left]\frac{1}{e^2}; \frac{1}{e}\right[\) tel que \(f(\alpha) = 0\)
    \end{enumerate}
    \item
    \begin{enumerate}[label=\alph*)]
        \item Montrer que \(y=x\) est une équation de la tangente \((T)\) à la courbe \((C)\) au point \(A(1;1)\)
        \item Vérifier que : \(f(x) - x = (\ln x + 1)h(x)\) pour tout \(x\) de \(]0; +\infty [\)
        \item En déduire que \((C)\) coupe la tangente \((T)\) en deux points dont on déterminera les coordonnées.
        \item Étudier la position relative de la courbe \((C)\) et la droite \((T)\) sur l’intervalle \(\left]\frac{1}{e}; 1\right[\)
    \end{enumerate}
    \item Tracer la courbe \((C)\) et la droite \((T)\) dans le repère orthonormé \((O; \vec{i}; \vec{j})\) (unité : 2 cm)
    (On admettra que la courbe \((C)\) possède un seul point d’inflexion dont l’abscisse est comprise entre \(\frac{1}{e}\) et 1)
\end{enumerate}
\noindent\textbf{IV. }
Soit \((U_n)\) la suite définie par :
~\(U_0 = \frac{1}{2} \quad \text{et} \quad U_{n+1} = f(U_n) \quad \text{pour tout } n \in \mathbb{N}\)~
\begin{enumerate}
    \item Montrer que : \(\frac{1}{e} \leq U_n \leq 1\) pour tout \(n \in \mathbb{N}\)
    \item Montrer que la suite \((U_n)\) est croissante.
    \item Montrer que la suite \((U_n)\) est convergente et déterminer sa limite.
\end{enumerate}
}





% Bottom message
\vspace*{-0.75cm}
\begin{center}
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    \RL{\arabicfont
    ﴿بِسْمِ ٱللَّهِ ٱلرَّحْمَـٰنِ ٱلرَّحِيمِ الٓر ۚ كِتَـٰبٌ أَنزَلْنَـٰهُ إِلَيْكَ لِتُخْرِجَ ٱلنَّاسَ مِنَ ٱلظُّلُمَـٰتِ إِلَى ٱلنُّورِ بِإِذْنِ رَبِّهِمْ إِلَىٰ صِرَٰطِ ٱلْعَزِيزِ ٱلْحَمِيدِ﴾ (إبراهيم 1)
    }%
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\end{document}