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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%	Cadre jaune
\textbf{Nom : }$\dots\dots\dots\dots\dots$ \hfill \textbf{Pr\'enom : }$\dots\dots\dots\dots\dots$  \hfill Classe :$\dots\dots$\\
\psframebox[linestyle=solid,shadow=true,framearc=0.5,fillcolor=vert!30]{
\begin{minipage}{17cm}
\begin{center}
\vspace{1mm}
\textsc{\textbf{ Interrogation n$^{\circ}15$ }}
\end{center}
\vspace{-3mm}
\end{minipage}
}
\setcounter{section}{1}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%	 Interro
\begin{center}
\textit{\textbf{On prendra soin de coller le sujet sur la copie. La note tiendra compte de la qualité de la rédaction et des justifications.}}
\end{center}


\medskip
 
 
 \Exoi{\hfill (4 points)\\
 \begin{enumerate}
 \item Montrer que pour tout réel $x$ on a :
 $$ \sqrt{x^2+1}-x>0\qquad \text{et} \qquad \sqrt{x^2+1}+x>0$$
 \item Montrer que pour tout réel $x$ on a :
 $$ \left(\sqrt{x^2+1}-x\right)\left(\sqrt{x^2+1}+x\right)=1$$
 \item Montrer que pour tout réel $x$ on a :
 $$ \ln\left(\sqrt{x^2+1}-x\right)+\ln \left(\sqrt{x^2+1}+x\right)=0$$
 \end{enumerate}
}


\Exoi{\hfill (6 points)\\
On considère la fonction $f$ définie sur $[0;+\infty[$ par :
$$ f(x)=x-\ln(1+x)$$
\begin{enumerate}
\item Calculer $f'(x)$ pour tout $x\in [0;+\infty[$.
\item Dresser le tableau de variation de la fonction $f$.
\item En déduire que pour tout réel $x\geq 0$ on a :
$$ \ln(1+x)\leq x$$
\item Déterminer la limite de $f$ en $+\infty$.\\
\textit{On posera $X=1+x$ afin d'utiliser les résultats connus du cours.} 
\end{enumerate}
}

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\vfill
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%	Cadre jaune
\textbf{Nom : }$\dots\dots\dots\dots\dots$ \hfill \textbf{Pr\'enom : }$\dots\dots\dots\dots\dots$  \hfill Classe :$\dots\dots$\\
\psframebox[linestyle=solid,shadow=true,framearc=0.5,fillcolor=vert!30]{
\begin{minipage}{17cm}
\begin{center}
\vspace{1mm}
\textsc{\textbf{ Interrogation n$^{\circ}15$ }}
\end{center}
\vspace{-3mm}
\end{minipage}
}
\setcounter{Exi}{0}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%	 Interro
\begin{center}
\textit{\textbf{On prendra soin de coller le sujet sur la copie. La note tiendra compte de la qualité de la rédaction et des justifications.}}
\end{center}
sdfghjklm

\Exoi{\hfill (4 points)\\
 \begin{enumerate}
 \item Dresser le tableau de signe de $x^2-2x-3$.
 \item En déduire une factorisation de $x^2-2x-3$.
 \item Montrer que pour $x>3$, on a :
 $$ \ln(x^2-2x-3)-2\ln(x+1)=\ln\left(\dfrac{x-3}{x+1}\right)$$
 \end{enumerate}
}

\Exoi{\hfill (6 points)\\
Soit $g$ la fonction définie pour tout nombre réel $x$ de l'intervalle $]0;+\infty[$ par :
$$g(x)=x-x\ln x $$
\begin{enumerate}
\item Déterminer les limites de la fonction $g$ en $0$ et $+\infty$.
\item Montrer que $g$ est dérivable sur $]0;+\infty[$ et calculer $g'(x)$.
\item Dresser le tableau de variations de la fonction $g$.
\item Résoudre l'équation $g(x)=0$ puis donner le tableau de signe de $g$.
\end{enumerate}}


\end{document}
