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\lhead{ \Large{COURS MPSI}}\chead{ \LARGE {B1.I. INÉGALITÉS}} \rhead {\Large{R. FERRÉOL 16/17}}
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\begin{document}
\fbox{1) \textbf{IN\'{E}GALITES }}
Rappel : $\mathbb{R=R}_{+}\cup \mathbb{R}_{-}=\mathbb{R}_{+}^{\ast }\cup
\mathbb{R}_{-}^{\ast }\cup \{0\}.$
DEF :
\begin{equation*}
\begin{tabular}{|l}
$a\leqslant b\Leftrightarrow b-a\in \mathbb{R}_{+}$ \\
$a0,\;\;a\leqslant
b\Leftrightarrow ac\leqslant bc$ \\ \hline
9. Inversion : $00 \\
0\text{ si }x=0 \\
-1\text{ si }x<0%
\end{array}%
\right. $ \\ \hline
$\left\vert x\right\vert =\left\{
\begin{array}{c}
x\text{ si }x\geqslant 0 \\
-x\;\text{si }x\leqslant 0%
\end{array}%
\right. $ \\ \hline
\end{tabular}%
\end{equation*}%
PROPRI\'{E}T\'{E}S :
\begin{equation*}
\begin{tabular}{|l|}
\hline
1) $x=x_{+}+x_{\_}$ \\ \hline
2) $\left\vert x\right\vert =\max \left( x,-x\right) =x_{+}-x_{\_}=x.\text{%
signe}(x)=\sqrt{x^{2}}$ \\ \hline
3) $\min \left( a,b\right) =\dfrac{a+b-\left\vert b-a\right\vert }{2},\max
\left( a,b\right) =\dfrac{a+b+\left\vert b-a\right\vert }{2}$ \\ \hline
4) $-a\leqslant x\leqslant a\Leftrightarrow \left\vert x\right\vert
\leqslant a$ \ ;\ $\left\vert x-x_{0}\right\vert \leqslant a\Leftrightarrow
x_{0}-a\leqslant x\leqslant x_{0}+a\Leftrightarrow x\in \left[
x_{0}-a,x_{0}+a\right] $ \\ \hline
5) $\left\vert xy\right\vert =\left\vert x\right\vert \left\vert
y\right\vert ,$ $\left\vert \dfrac{x}{y}\right\vert =\dfrac{\left\vert
x\right\vert }{\left\vert y\right\vert }$ \\ \hline
6) In\'{e}galit\'{e} triangulaire : $\left\vert x+y\right\vert \leqslant
\left\vert x\right\vert +\left\vert y\right\vert $ \\ \hline
7) In\'{e}galit\'{e} triangulaire gauche : $\left\vert \left\vert
x\right\vert -\left\vert y\right\vert \right\vert \leqslant $ $\left\vert
x+y\right\vert $ \\ \hline
\end{tabular}%
\end{equation*}%
D2
\bigskip
\fbox{\textbf{4) PARTIE ENTI\`{E}RE et fonctions apparent\'{e}es}.}
\bigskip
DEF :
La \textit{partie enti\`{e}re (inf\'{e}rieure) }de $x$ : $\left\lfloor
x\right\rfloor =$ E$(x)=\underline{\text{E}}\left( x\right) $ est le plus
grand entier inf\'{e}rieur ou \'{e}gal \`{a} $x.$\newline
(python : $floor(x)$ qui est un flottant, ou $int(x)$ qui est un entier$)$
La \textit{partie enti\`{e}re sup\'{e}rieure} de $x$ : $\left\lceil
x\right\rceil $ $=\overline{\text{E}}(x)$ est le plus petit entier sup\'{e}%
rieur ou \'{e}gal \`{a} $x$ (python : $ceil(x)).$
L'\textit{arrondi (entier)} de $x$ : rnd$(x)=\left[ x\right] $ est l'entier
le plus proche de $x$ ; s'il y en a deux, on prend le plus grand pour $%
x\geqslant 0,$ et le plus petit pour $x<0$ (python : $round(x)).$
La \textit{partie fractionnaire} de $x$ est la diff\'{e}rence entre $x$ et
sa partie enti\`{e}re : frac$\left( x\right) =\{x\}=x-\left\lfloor
x\right\rfloor .$
\bigskip
Attention, dans les calculatrices, et en particulier en python, "int"
arrondit vers 0 (int(-4.8)=-4) ; il ne correspond \`{a} la partie enti\`{e}%
re que pour les positifs.
PROPRI\'{E}T\'{E}S :
\begin{equation*}
\begin{tabular}{|l|}
\hline
$n=\left\lfloor x\right\rfloor \Leftrightarrow \left\{
\begin{array}{c}
n\in \mathbb{Z} \\
n\leqslant x