\documentclass[a4paper,12pt]{article}
\usepackage[left=1.00cm, right=1.00cm, top=2cm, bottom=1.50cm]{geometry}
\usepackage[utf8]{inputenc}
\usepackage{fontspec}
\usepackage{amsmath, amsfonts, amssymb, amsthm}
\usepackage{graphicx}
\usepackage{xcolor}
\usepackage{multicol}
\usepackage{enumitem}
\usepackage{mathrsfs}
\usepackage{colortbl}
\usepackage{tcolorbox,varwidth}

\tcbuselibrary{skins,breakable}
\usepackage{tikz}
\usetikzlibrary{calc}
\usetikzlibrary{shadows}
\usepackage{tabularx, array}
\usepackage{fancyhdr}
\usepackage{setspace}
\usepackage{hyperref}
\hypersetup{
    colorlinks=true,
    urlcolor=magenta
}

% French language support
    \usepackage[french]{babel}

% Bottom message in Arabic requires bidi package
    \usepackage{bidi}
    \newfontfamily\arabicfont[Script=Arabic,Scale=1.1]{Amiri}

% Two-page layout (conditionally included)







% --- Colors ---
\definecolor{maroon}{cmyk}{0,0.87,0.68,0.32}
\definecolor{pBlue}{RGB}{102, 53, 153}

\definecolor{myblue}{HTML}{1D2A6E}
\definecolor{myred}{HTML}{A97E77}
\definecolor{mybrown}{HTML}{B35823}
\definecolor{gold}{HTML}{B48920}
\definecolor{myorange}{HTML}{ED8C2B}
\definecolor{col}{RGB}{45, 136, 119}
\definecolor{MainRed}{rgb}{.8, .1, .1}
\definecolor{winered}{rgb}{0.5,0,0}
\definecolor{JungleGreen}{HTML}{29AB87}

\definecolor{myblue3}{RGB}{36, 57, 126}
\definecolor{mygray}{RGB}{112,121,139}
\definecolor{LemonGlacier}{RGB}{253,255,0}
\colorlet{myblue3}{red!75!black}

\definecolor{lightgray}{gray}{0.6}
\definecolor{arrowblue}{RGB}{0,140,180}
\definecolor{bannerblue}{RGB}{210,240,240}

% --- 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 libre N\s{$\circ$}2 -- Par: Pr. MOHAMED JOUAHRI}
\def\schoolname{\textbf{Lycée :} Lycée : Taghzirt}
\def\academicyear{2025/2026}
\def\subject{Mathématiques}
\def\duration{2h}
\def\secondtitle{\small{(Fonctions primitives,Suites \& Etude des fonctions)}}
\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~~}
\def\bottommsg{\\vspace*{-0.5cm}\n\\begin{center}\n  \\RL{\\arabicfont\n  ﴿بِسْمِ ٱللَّهِ ٱلرَّحْمَـٰنِ ٱلرَّحِيمِ طه (1)•  مَآ أَنزَلْنَا عَلَيْكَ ٱلْقُرْءَانَ لِتَشْقَىٰٓ (2)•  إِلَّا تَذْكِرَةً لِّمَن يَخْشَىٰ (3)•  تَنزِيلًا مِّمَّنْ خَلَقَ ٱلْأَرْضَ وَٱلسَّمَـٰوَٰتِ ٱلْعُلَى (4)•  ٱلرَّحْمَـٰنُ عَلَى ٱلْعَرْشِ ٱسْتَوَىٰ (5)•  لَهُۥ مَا فِى ٱلسَّمَـٰوَٰتِ وَمَا فِى ٱلْأَرْضِ وَمَا بَيْنَهُمَا وَمَا تَحْتَ ٱلثَّرَىٰ (6)•  وَإِن تَجْهَرْ بِٱلْقَوْلِ فَإِنَّهُۥ يَعْلَمُ ٱلسِّرَّ وَأَخْفَى (7)•  ٱللَّهُ لَآ إِلَـٰهَ إِلَّا هُوَ ۖ لَهُ ٱلْأَسْمَآءُ ٱلْحُسْنَىٰ (8)• ﴾ (طه الآيات 1-8)\n  }\n\\end{center}\n}
\def\direction{ltr}

% --- Exam-specific definitions ---

% Exam-specific settings and custom definitions
% This file is generated in the current working directory
% Edit this file to customize your exam settings

% Define column type for centered cells
\newcolumntype{Y}{>{\centering\arraybackslash}X}
\newcolumntype{M}[1]{@{}>{\centering\arraybackslash}m{\#1}@{}}

\newcommand{\tb}{\tikz[baseline=-0.6ex]{\fill (0,0) circle (2pt);}~}

\newcommand{\ccc}[1]{
    \begin{tikzpicture}[overlay, remember picture]
        \node[circle, inner sep=3pt, draw=black, outer sep=0pt] at (0.5,0.2) {\#1};
    \end{tikzpicture}
}

% Document spacing
\setstretch{1.3}

% Math display style
\everymath{\displaystyle}

%\AtBeginDocument{\fontsize{15}{17}\selectfont}

%\newcommand{\fff}[1]{\makebox[#1cm]{\dotfill}}

\SetEnumitemKey{tight}{
    leftmargin=*,
    itemsep=0pt,
    topsep=0pt,
    parsep=0pt,
    partopsep=0pt
}

\newcommand{\s}{\textsuperscript}

% 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)
% \usepackage{siunitx} % For SI units
% \usepackage{chemformula} % For chemical formulas
% \usepackage{circuitikz} % For circuit diagrams

% Page layout adjustments
% \setlength{\parindent}{0pt}
% \setlength{\parskip}{1em}

% Custom colors for this exam
% \definecolor{myblue}{RGB}{0,100,200}
% \definecolor{mygreen}{RGB}{0,150,0}

% ----------------------------------------------------------------------
% ADD YOUR CUSTOM DEFINITIONS BELOW THIS LINE
% ----------------------------------------------------------------------

% Example:
\newcommand{\dif}{\mathrm{d}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\C}{\mathbb{C}}



% Exercise Theme 6: Maroon title bar with triangles
\newtcolorbox{exothemesix}[2][]{%
  before=\centering,
  after={},
  before skip=0pt,
  after skip=0pt,
  boxsep=0pt,
  left=4pt,
 right=2pt,
 breakable,
  enhanced,
  skin=enhancedlast jigsaw,
  attach boxed title to top left={xshift=-4mm,yshift=-0.5mm},
  fonttitle=\bfseries\sffamily,
  varwidth boxed title=0.7\linewidth,
  colbacktitle=gray!45!white,
  colframe=red!50!black,
  interior style={top color=gray!30!white,bottom color=gray!10!white},
  boxed title style={empty,arc=0pt,outer arc=0pt,boxrule=0pt},
  underlay boxed title={
    \fill[maroon] (title.north west) -- (title.north east)
    -- +(\tcboxedtitleheight-1mm,-\tcboxedtitleheight+1mm)
    -- ([xshift=4mm,yshift=0.5mm]frame.north east) -- +(0mm,-1mm)
    -- (title.south west) -- cycle;
    \fill[maroon!70!white!50!black] ([yshift=-0.5mm]frame.north west)
    -- +(-0.4,0) -- +(0,-0.3) -- cycle;
    \fill[maroon!45!white!50!black] ([yshift=-0.5mm]frame.north east)
    -- +(0,-0.3) -- +(0.4,0) -- cycle;
  },
  title=#2,
  #1
}

\newcommand{\printexo}[3]{%
  \if\relax\detokenize{\#2}\relax
    \def\fulltitle{Exercice #1}%
  \else
    \def\fulltitle{Exercice #1~#2}%
  \fi
  \begin{exothemesix}[]{\fulltitle}#3\end{exothemesix}%
  \vspace{0.2cm}%
}

% --- Custom Header (fr) ---
% Custom header based on default header style 9
% Language: fr
\newcommand{\printheadnine}{%
  \arrayrulecolor{lightgray}
  \begin{tabular}{m{0.17\textwidth} m{0.61\textwidth} m{0.18\textwidth}}
      \textbf{\classname} & \centering \textbf{\examtitle}
      & \ddate \\
      \wsite & \centering \large\textbf{\secondtitle}
      &\hfill
        \begin{tabular}{|c}
          \hline
          \textbf{~~\professor}
        \end{tabular}\\
      \hline
  \end{tabular}
  \arrayrulecolor{black}
}

\fancyhf{}
\renewcommand{\headrulewidth}{0pt}
\renewcommand{\footrulewidth}{0pt}
\setlength{\headheight}{47pt}
\setlength{\headsep}{-8pt}
\fancyhead[C]{%
  \printheadnine
}%
\pagestyle{fancy}

\begin{document}



% Exercise 1 (originally 1)
\printexo{1}{}{
%\vspace*{-0.75cm}
\begin{enumerate}[label=\arabic*., tight]
    \item Dans chacun des cas suivants, déterminer les primitives de la fonction \( f \) sur l’intervalle \( I = \mathbb{R} \):\\
    ~$
    f(x) = x^2 + 3x\sqrt{x^2 + 2} ; \quad f(x) = \frac{x}{\sqrt{x^2 + 10}} ; \quad f(x) = 3x^2(x^3 + 1)^5 + \sin(2x)
    $~
    \item Soit \( g \) la fonction numérique définie sur l’intervalle \( I = ]1, +\infty[ \) par :
    ~$
    g(x) = \frac{2x + 5}{(x - 1)^3}.
    $~
    \begin{enumerate}[label=\alph*)]
        \item Déterminer les réels \( a \) et \( b \) tels que :
        ~$
        (\forall x \in I) \quad g(x) = \frac{a}{(x - 1)^2} + \frac{b}{(x - 1)^3}.
        $~
        \item En déduire la primitive de la fonction \( g \) sur l’intervalle \( I \) qui s’annule en 0.
    \end{enumerate}
\end{enumerate}
}

% Exercise 2 (originally 2)
\printexo{2}{}{
Soit \((U_n)\) la suite numérique définie par :
~$
U_0 = 0 \quad \text{et} \quad U_{n+1} = \frac{2U_n + 1}{U_n + 2} ; \quad (\forall n \in \mathbb{N})
$~

\begin{enumerate}[label=\arabic*., tight]
    \item Montrer que :
    ~$
    (\forall n \in \mathbb{N}) : \quad 0 \leq U_n < 1.
    $~
    \item Montrer que la suite \((U_n)\) est croissante, en déduire qu’elle est convergente.
    \item Montrer que :
    ~$
    (\forall n \in \mathbb{N}) \quad 1 - U_{n+1} \leq \frac{1}{2}(1 - U_n).
    $~
    \item En déduire que :
    ~$
    (\forall n \in \mathbb{N}) \quad 1 - U_n \leq \left(\frac{1}{2}\right)^n, \text{ puis calculer } \lim U_n.
    $~
    \item Soit \( n \in \mathbb{N} \) on pose :
    ~$
    V_n = \frac{U_n - 1}{U_n + 1}.
    $~
    \begin{enumerate}[label=\alph*)]
        \item Montrer que \((V_n)\) est géométrique de raison \( q = \frac{1}{3} \).
        \item Déterminer \( V_n \) en fonction de \( n \) puis \( U_n \) en fonction de \( n \).
    \end{enumerate}
\end{enumerate}
}

% Exercise 3 (originally 3)
\printexo{3}{}{
\textbf{Partie A}\\
Soit \( f \) la fonction numérique définie sur \( \mathbb{R}_+ \) par :
~$
f(x) = 4x\sqrt{x} - 3x^2.
$~

\begin{enumerate}[label=\arabic*., tight]
    \item Étudier la dérivabilité de \( f \) à droite en 0 puis interpréter graphiquement le résultat obtenu.
    \item  {\fontsize{11}{13}\selectfont Calculer \( \lim_{x \to +\infty} f(x) \) puis \( \lim_{x \to +\infty} \frac{f(x)}{x} \)
      et déduire la nature de la branche infinie de
      \((\mathscr C_f)\)  au voisinage de \( +\infty \).}
    \item  Montrer que :
    ~$
    (\forall x \in \mathbb{R}_+^*) \quad f'(x) = 6\sqrt{x}\left(1 - \sqrt{x}\right),
    $~
     {\fontsize{11}{13}\selectfont puis étudier les variations de la fonction} \( f \) sur \( \mathbb{R}_+ \).
    \item
    \begin{enumerate}[label=\alph*)]
        \item Vérifier que :
        ~$
        (\forall x \in \mathbb{R}_+^*) \quad f(x) - x = x\left(\sqrt{x} - 1\right)\left(1 - 3\sqrt{x}\right).
        $~
        \item Étudier la position relative de \((\mathscr C_f)\) et la droite \((D)\) d’équation \( y = x \).
    \end{enumerate}
    \item Montrer que :
    ~$
    (\forall x \in \mathbb{R}_+^*) \quad f''(x) = \frac{3\left(1 - 2\sqrt{x}\right)}{\sqrt{x}},
    $~
    puis étudier la concavité de \((\mathscr C_f)\).
    \item Déterminer les points d’intersections de \((\mathscr C_f)\) et l’axe des abscisses, puis construire \((\mathscr C_f)\).
\end{enumerate}
\textbf{Partie B}\\
Soit \((U_n)\) la suite numérique définie par :
~$
U_0 = \frac{4}{9} \quad \text{et} \quad (\forall n \in \mathbb{N}) \quad U_{n+1} = f(U_n).
$~

\begin{enumerate}[label=\arabic*., tight]
    \item Montrer que :
    ~$
    (\forall n \in \mathbb{N}) : \quad \frac{1}{9} \leq U_n \leq 1.
    $~
    \item Montrer que la suite \((U_n)\) est croissante puis déterminer \( \lim U_n \).
\end{enumerate}
}






% Bottom message
\vspace*{-0.5cm}
\begin{center}
  \RL{\arabicfont
  ﴿بِسْمِ ٱللَّهِ ٱلرَّحْمَـٰنِ ٱلرَّحِيمِ طه (1)•  مَآ أَنزَلْنَا عَلَيْكَ ٱلْقُرْءَانَ لِتَشْقَىٰٓ (2)•  إِلَّا تَذْكِرَةً لِّمَن يَخْشَىٰ (3)•  تَنزِيلًا مِّمَّنْ خَلَقَ ٱلْأَرْضَ وَٱلسَّمَـٰوَٰتِ ٱلْعُلَى (4)•  ٱلرَّحْمَـٰنُ عَلَى ٱلْعَرْشِ ٱسْتَوَىٰ (5)•  لَهُۥ مَا فِى ٱلسَّمَـٰوَٰتِ وَمَا فِى ٱلْأَرْضِ وَمَا بَيْنَهُمَا وَمَا تَحْتَ ٱلثَّرَىٰ (6)•  وَإِن تَجْهَرْ بِٱلْقَوْلِ فَإِنَّهُۥ يَعْلَمُ ٱلسِّرَّ وَأَخْفَى (7)•  ٱللَّهُ لَآ إِلَـٰهَ إِلَّا هُوَ ۖ لَهُ ٱلْأَسْمَآءُ ٱلْحُسْنَىٰ (8)• ﴾ (طه الآيات 1-8)
  }
\end{center}


\end{document}