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\begin{document}

%%%%%%%%%%b{gedeonInfo}%%%%%%%%%
%%--Document N° 119902713265529--
%%--Application de la dérivée--
%%%%%%%%%%e{gedeonInfo}%%%%%%%%%


\begin{exercise}
Dans chacun des cas suivants, donner l'allure possible de la courbe
$\mathcal{C}_{f}$, sachant que $f$ est une fonction d\'{e}rivable sur $%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
$ v\'{e}rifiant les conditions donn\'{e}es:

\begin{enumerate}
\item $f(-2)=3\quad;\quad f(0)=1\quad;\quad f(-4)=0\quad;\quad f^{\prime
}(-2)=f^{\prime}(0)=0\quad;\quad f^{\prime}(x)>0$ sur $]-\infty;-2[$ et sur
$]0;+\infty\lbrack;$

\item $\forall n\in%
%TCIMACRO{\U{2124} }%
%BeginExpansion
\mathbb{Z}
%EndExpansion
,\quad f(n)=n$ et $f^{\prime}(n)=0\quad;\quad\forall x\notin%
%TCIMACRO{\U{2124} }%
%BeginExpansion
\mathbb{Z}
%EndExpansion
,\quad f^{\prime}(x)>0;$

\item $f(0)=3\quad;\quad f(2)=f^{\prime}(2)=0\quad;\quad f^{\prime}(x)<0$ sur
$]0;2[\quad;\quad f^{\prime}(x)>0$ sur $]2;+\infty\lbrack\quad;\quad f$ est paire.
\end{enumerate}
\end{exercise}

\begin{exercise}
Les courbes de la page \pageref{PgeCourbes} repr\'{e}sentent quatre fonctions
(courbes $\mathcal{C}_{1},$ $\mathcal{C}_{2},$ $\mathcal{C}_{3}$ et
$\mathcal{C}_{4}$) et leurs fonctions d\'{e}riv\'{e}es (courbes $\mathcal{C}%
_{1}^{\prime},$ $\mathcal{C}_{2}^{\prime},$ $\mathcal{C}_{3}^{\prime}$ et
$\mathcal{C}_{4}^{\prime}$) dans un ordre quelconque. Observer attentivement
ces courbes et associer \`{a} chaque fonction sa courbe d\'{e}riv\'{e}e
\textbf{en pr\'{e}cisant bri\`{e}vement votre d\'{e}marche}.
\end{exercise}

\begin{exercise}
On pose, pour $n\in\{2;3;4;...\},$ $f(x)=x^{n}.$

\begin{enumerate}
\item R\'{e}soudre l'\'{e}quation $f^{\prime}(x)=0;$

\item Pour quelles valeurs de $n$ $f$ admet-elle un extremum local en 0?
\end{enumerate}
\end{exercise}

\begin{exercise}
On pose, pour $a\in%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
,$ $f(x)=x^{3}+ax.$

\begin{enumerate}
\item R\'{e}soudre l'\'{e}quation $f^{\prime}(x)=0;$

\item Pour quelles valeurs de $a$ $f$ admet-elle, au moins, un extremum local?
\end{enumerate}
\end{exercise}

\begin{exercise}
On pose, pour $a\in%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
,$ $f(x)=x^{3}+ax^2+x+1.$

\begin{enumerate}
\item Déterminer, suivants les valeurs de $a$, \textbf{le nombre} de solution de
  l'équation $f^{\prime}(x)=0$.

\item Pour quelles valeurs de $a$ $f$ admet-elle, au moins, un
  extremum local?\\
  \textit{On aura soin de préciser son raisonnement}.
\end{enumerate}
\end{exercise}


\begin{exercise}
Dans chacun des cas suivants, \'{e}tudier les variations de $f$ sur l'ensemble
consid\'{e}r\'{e}, et pr\'{e}ciser les extremums locaux:%

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\begin{enumerate}
\item $f(x)=4x-3+1/x$ sur $%
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\mathbb{R}
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^{\star};$

\item $f(x)=\sin^{2}x$ sur $]-\pi;\pi\lbrack;$

\item $f(x)=\sin(5x+\pi/4)$ sur $%
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\mathbb{R}
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;$

\emph{Imdication: }$f$ est $\tfrac{2\pi}{5}$-p\'{e}riodique.

\item $f(x)=\dfrac{1-\cos x}{1+\cos x}$ sur $]-\pi;\pi\lbrack.$
\end{enumerate}

%

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\end{exercise}

\begin{exercise}
\'{E}tudier les variations de $f\,:\,x\longmapsto\cos^{2}x$ sur
$]-\pi;\pi\lbrack$.\\
\textit{On sera amené à construire un tableau de signe}.
\end{exercise}


\begin{exercise}
D\'{e}montrer l'in\'{e}galit\'{e} de \textsc{Bernoulli}: $\forall x\in%
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\mathbb{R}
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^{+}$ et $\forall n\in%
%TCIMACRO{\U{2115}^\star }%
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\mathbb{N}^\star
%EndExpansion
$ on a $x^{n}-1\geq n(x-1)$.
\end{exercise}

\begin{exercise}
Un r\'{e}cipient, parall\'{e}l\'{e}pip\'{e}de rectangle, dont la base est un
carr\'{e} de $20%
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\operatorname{cm}%
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$ de cot\'{e} (et de hauteur assez grande) est rempli d'eau jusqu'\`{a} une
hauteur de $10%
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\operatorname{cm}%
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.$

Peut-on d\'{e}poser sur le fond une bille sph\'{e}rique qui affleure la
surface de l'eau? Dans l'affirmative, donner un encadrement d'amplitude
$10^{-3}$ du rayon de la bille.\newpage
\end{exercise}

\begin{exercise}
On dispose d'une feuille de carton rectangulaire de côtés 20 et 15. On
enlève dans chaque coin un carré de côté $x$ afin d'obtenir après
pliage des bords une boite (sans couvercle) en forme de pavé droit.\\
Quelle doit être la valeur de $x$ pour que le volume de la boite soit
maximal ?
\begin{figure}[h]
  \centering
\includegraphics{Include/figure.eps}  
  \caption{Plier suivant les pointillés}
  \label{feuille_irie}
\end{figure}
\end{exercise}


\begin{center}%
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[c]{lll}%
$%
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$%
\end{tabular}
\label{PgeCourbes}
\end{center}


\end{document}
