\begin{enumerate}
\item $f(x)>g(x)$ : il faut regarder les portions de courbes de $\mathcal{C}_f$ qui sont au-dessus de la courbe $\mathcal{C}_g$.
\begin{center}
\begin{tikzpicture}[scale=0.75,every node/.style={scale=0.75}]
\draw[dashed,gray!50] (-5,-5) grid (5,5);
\draw[->,>=latex] (-5,0)--(5,0);
\draw[->,>=latex] (0,-5)--(0,5);
\node[below left] at (0,0) {0};
\foreach\x in {-4,-3,-2,-1,1,2,3,4}
{
\draw (\x,0.2) -- (\x,-0.2) node[below] {$\x$};
\draw (0.2,\x) -- (-0.2,\x) node[left] {$\x$};
}
\clip (-5,-5) rectangle (5,5);
\draw[red] plot[domain=-5:5,samples=100] (\x,{0.5*(\x-3)*(\x-1)*(\x+2)});
\draw[violet] plot[domain=-5:5,samples=100] (\x,{-\x*\x/3+3});
\node[red] at (4,4.5) {$\mathcal{C}_f$};
\node[violet] at (4,-4.5) {$\mathcal{C}_g$};
%
\draw[red,very thick] plot[domain=-1.65:0,samples=100] (\x,{0.5*(\x-3)*(\x-1)*(\x+2)});
\draw[red,thick] plot[domain=3:4,samples=100] (\x,{0.5*(\x-3)*(\x-1)*(\x+2)});
\draw[red,dashed] (-1.65, {0.5*(-1.65-3)*(-1.65-1)*(-1.65+2)}) -- (-1.65,0);
\draw[very thick,red] (-1.65,0) -- (0,0);
\draw[very thick,red] (3,0) -- (5,0);
\end{tikzpicture}
\end{center}

Les intervalles de $x$ qui correspondent à ces \og morceaux \fg{} de courbes sont \intervOO{-1,6}{0} et \intervOF{3}{5} (les bornes ne sont pas incluses -- sauf 5 -- car l'inégalité doit être stricte : $f(x)>g(x)$. L'ensemble solution est donc :
\[ \boxed{S = \intervOO{-1,6}{0}\cup\intervOF{3}{5}} \]

\newpage

\item $f(x)\geqslant0$. Il faut regarder les portions de la courbe $\mathcal{C}_f$ qui sont au-dessus de l'axe des abscisses.
\begin{center}
\begin{tikzpicture}[scale=0.75,every node/.style={scale=0.75}]
\draw[dashed,gray!50] (-5,-5) grid (5,5);
\draw[->,>=latex] (-5,0)--(5,0);
\draw[->,>=latex] (0,-5)--(0,5);
\node[below left] at (0,0) {0};
\foreach\x in {-4,-3,-2,-1,1,2,3,4}
{
\draw (\x,0.2) -- (\x,-0.2) node[below] {$\x$};
\draw (0.2,\x) -- (-0.2,\x) node[left] {$\x$};
}
\clip (-5,-5) rectangle (5,5);
\draw[red] plot[domain=-5:5,samples=100] (\x,{0.5*(\x-3)*(\x-1)*(\x+2)});
\draw[violet,dotted] plot[domain=-5:5,samples=100] (\x,{-\x*\x/3+3});
\node[red] at (4,4.5) {$\mathcal{C}_f$};
\node[violet] at (4,-4.5) {$\mathcal{C}_g$};
%
\draw[red,very thick] plot[domain=-2:1,samples=100] (\x,{0.5*(\x-3)*(\x-1)*(\x+2)});
\draw[red,thick] plot[domain=3:4,samples=100] (\x,{0.5*(\x-3)*(\x-1)*(\x+2)});
\draw[very thick,red] (-2,0) -- (1,0);
\draw[very thick,red] (3,0) -- (5,0);
\end{tikzpicture}
\end{center}
L'ensemble solution est :
\[ \boxed{S = \intervFF{-2}{1}\cup\intervFF{3}{5}} \]

\item $g(x)\leqslant0$. Il faut regarder ici les morceaux de la courbe $\mathcal{C}_g$ sous l'axe des abscisses.
\begin{center}
\begin{tikzpicture}[scale=0.75,every node/.style={scale=0.75}]
\draw[dashed,gray!50] (-5,-5) grid (5,5);
\draw[->,>=latex] (-5,0)--(5,0);
\draw[->,>=latex] (0,-5)--(0,5);
\node[below left] at (0,0) {0};
\foreach\x in {-4,-3,-2,-1,1,2,3,4}
{
\draw (\x,0.2) -- (\x,-0.2) node[below] {$\x$};
\draw (0.2,\x) -- (-0.2,\x) node[left] {$\x$};
}
\clip (-5,-5) rectangle (5,5);
\draw[red,dotted] plot[domain=-5:5,samples=100] (\x,{0.5*(\x-3)*(\x-1)*(\x+2)});
\draw[violet] plot[domain=-5:5,samples=100] (\x,{-\x*\x/3+3});
\node[red] at (4,4.5) {$\mathcal{C}_f$};
\node[violet] at (4,-4.5) {$\mathcal{C}_g$};
%
\draw[violet,very thick] plot[domain=-5:-3,samples=100] (\x,{-\x*\x/3+3});
\draw[violet,very thick] plot[domain=3:5,samples=100] (\x,{-\x*\x/3+3});
\draw[very thick,violet] (-5,0) -- (-3,0);
\draw[very thick,violet] (3,0) -- (5,0);
\end{tikzpicture}
\end{center}
L'ensemble solution est :
\[ \boxed{S = \intervFF{-5}{-3}\cup\intervFF{3}{5}} \]
\end{enumerate}
\newpage