\newpage
\exercice[style=R,N=3,verif]{Lectures graphiques de f, f' et f''}

Soit $f$ une fonction deux fois d\'erivable sur \R. On note $f'$ sa d\'eriv\'ee et $f{''}$ sa d\'eriv\'ee seconde.

La  courbe repr\'esentative de la fonction d\'eriv\'ee not\'ee $\mathcal{C}_{f'}$ est donn\'ee ci-dessous. 

La droite $\cal T$ est tangente \`a la courbe $\mathcal{C}_{f'}$ au point d'abscisse 0.
\begin{center}
\begin{tikzpicture}[>=stealth',scale=0.9]
\draw[gray,dashed] (-3,-3) grid (4,2);
\draw[->] (-3,0) -- (4,0);
\draw[->] (0,-3) -- (0,2);
\foreach\x in {-3,-2,-1,1,2,3}
{
\draw (\x,0.1) -- (\x,-0.1) node[below] {$\x$};
}
\foreach\y in {-3,-2,-1,1,2}
{
\draw (0.1,\y) -- (-0.1,\y) node[left] {$\y$};
}
\clip (-3,-3) rectangle (4,2);
\draw[thick,violet] plot[domain=-3:4,samples=100] (\x,{0.5*(\x+2)*exp(-\x)});
%
\draw[green!50!black] plot[domain=-3:4] (\x,{-0.5*\x+1});
\node[green!50!black] at (3.75,-1.4) {$\cal T$};
\end{tikzpicture}
\end{center}
\vspace*{-2em}
\begin{enumerate}
	\item Par lecture graphique :
\begin{enumerate}
	\item R\'esoudre $f'(x)=0$.
	\item R\'esoudre $f{''}(x)=0$.
	\item D\'eterminer $f{''}(0)$.
\end{enumerate}
\item Une des quatre courbes $\mathcal{C}_1$, $\mathcal{C}_2$, $\mathcal{C}_3$ et $\mathcal{C}_4$ ci-dessous est la courbe repr\'esentative de la fonction $f$ et une autre est celle de  la d\'eriv\'ee seconde $f{''}$.
\begin{center}
\begin{tikzpicture}[>=stealth',scale=0.9]
\draw[gray,dashed] (-3,-3) grid (4,2);
\draw[->] (-3,0) -- (4,0);
\draw[->] (0,-3) -- (0,2);
\foreach\x in {-3,-2,-1,1,2,3}
{
\draw (\x,0.1) -- (\x,-0.1) node[below] {$\x$};
}
\foreach\y in {-3,-2,-1,1,2}
{
\draw (0.1,\y) -- (-0.1,\y) node[left] {$\y$};
}
\clip (-3,-3) rectangle (4,2);
\draw[thick,violet] plot[domain=-3:4,samples=100] (\x,{-.5*(\x+3)*exp(-\x)+0.5*exp(2)-1});
\end{tikzpicture}
\hspace*{1cm}
\begin{tikzpicture}[>=stealth',scale=0.9]
\draw[gray,dashed] (-3,-3) grid (4,2);
\draw[->] (-3,0) -- (4,0);
\draw[->] (0,-3) -- (0,2);
\foreach\x in {-3,-2,-1,1,2,3}
{
\draw (\x,0.1) -- (\x,-0.1) node[below] {$\x$};
}
\foreach\y in {-3,-2,-1,1,2}
{
\draw (0.1,\y) -- (-0.1,\y) node[left] {$\y$};
}
\clip (-3,-3) rectangle (4,2);
\draw[thick,violet] plot[domain=-3:4,samples=100] (\x,{.5*(\x+3)*exp(-\x)-0.5*exp(2)+1});
\end{tikzpicture}

\begin{tikzpicture}[>=stealth',scale=0.9]
\draw[gray,dashed] (-3,-3) grid (4,2);
\draw[->] (-3,0) -- (4,0);
\draw[->] (0,-3) -- (0,2);
\foreach\x in {-3,-2,-1,1,2,3}
{
\draw (\x,0.1) -- (\x,-0.1) node[below] {$\x$};
}
\foreach\y in {-3,-2,-1,1,2}
{
\draw (0.1,\y) -- (-0.1,\y) node[left] {$\y$};
}
\clip (-3,-3) rectangle (4,2);
\draw[thick,violet] plot[domain=-3:4,samples=100] (\x,{-.5*(\x+1)*exp(-\x)});
\end{tikzpicture}
\hspace*{1cm}
\begin{tikzpicture}[>=stealth',scale=0.9]
\draw[gray,dashed] (-3,-3) grid (4,2);
\draw[->] (-3,0) -- (4,0);
\draw[->] (0,-3) -- (0,2);
\foreach\x in {-3,-2,-1,1,2,3}
{
\draw (\x,0.1) -- (\x,-0.1) node[below] {$\x$};
}
\foreach\y in {-3,-2,-1,1,2}
{
\draw (0.1,\y) -- (-0.1,\y) node[left] {$\y$};
}
\clip (-3,-3) rectangle (4,2);
\draw[thick,violet] plot[domain=-3:4,samples=100] (\x,{.5*(\x+1)*exp(-\x)});
\end{tikzpicture}
\end{center}
\begin{enumerate}
	\item D\'eterminer la courbe qui repr\'esente $f$ et celle qui repr\'esente la d\'eriv\'ee seconde $f{''}$.
	\item D\'eterminer les intervalles sur lesquels $f$ est convexe ou concave.
	\item La courbe repr\'esentative de la fonction $f$ admet-elle un point d'inflexion ?
\end{enumerate}
\end{enumerate}
