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\begin{document}

 Maths cc~~~~~~~~~~~~~~~~{\Huge {\bf 
                      ~Exo-type 5 }}  \hfill x/x/x ~Deug MIAS\\ %

		     \vspace{0.6cm}
\noindent
Nom:\\
Documents:\\
Calculette:
		     \vspace{0.6cm}

On consid\`ere une fonction $f$ dont voici le TV. 

$$\begin{array}{|c|rcccccl|}
\hline
x& -\infty&&1&&3&&+\infty\\
\hline
f(x)&-\infty&\nearrow&-1&\nearrow&6&\nearrow&+\infty\\
\hline
f'(x)&2&\nearrow&3&\nearrow&4&\nearrow&+\infty\\
\hline
\end {array}
$$


a) Faites un croquis du graphe de $f$ avec la diagonale.

		     \vspace{01cm}

b) On consid\`ere une suite $u$ v\'erifiant $u_{n+1}=f(u_n)$.
Remplissez le tableau suivant concernant la suite $u$ selon la valeur de
$u_0$:

		     \vspace{1cm}

$$
\begin{array}{|c|c|c|c|c|c|}
\hline
u_0=&$1000$&$-1000$&$2$&$1,2$&$1,8$\\ \hline
\text {croissante?}&~~~~~~~~~~~~~~~&~~~~~~~~~~~~~~~&~~~~~~~~~~~~~~~
&~~~~~~~~~~~~~~~&~~~~~~~~~~~~~~~
\\ \hline
\text {d\'ecroissante?}&&&&&\\ \hline
\text {convergente?}&&&&&\\ \hline
\text {born\'ee?}&&&&&\\ \hline
\text {monotone?}&&&&&\\ \hline
\text {stationnaire?}&&&&&\\ \hline
sup&&&&&\\ \hline
inf&&&&&\\ \hline
max=sup?&&&&&\\ \hline
min=inf?&&&&&\\ \hline
lim&&&&&\\ \hline
\end {array}
$$

		     \vspace{0.6cm}


c) M\^emes questions pour la fonction $f$ dont voici le TV: 

$$\begin{array}{|c|rcccccl|}
\hline
x& -\infty&&-5&&6&&+\infty\\
\hline
f(x)&-\infty&\nearrow&-4&\nearrow&5&\nearrow&+\infty\\
\hline
f'(x)&0&\nearrow&1&\searrow&\frac {\pi}{4}&\nearrow&1\\
\hline
\end {array}
$$

et 

$$
\begin{array}{|c|c|c|c|c|c|}
\hline
u_0=
&~~~~~1000~~~~~~&~~~~~-1000~~~~~&~~~~~~~-4~~~~~~~
&~~~~~~5~~~~~~~~&~~~~~~~1~~~~~~~
\\ \hline
\end {array}
$$

	\end{document}


\begin{tabular}{|l|c|c|c|c|}
\hline
&$I$&$J$&$K$&$L$\\ \hline
croissante?&~~~~~~~~~~~~~~~~~~~~~~~~~&~~~~~~~~~~~~~~~~~~~~~~~~~~
&~~~~~~~~~~~~~~~~~~~~~~~~~~&~~~~~~~~~~~~~~~~~~~~~~~~~~\\ \hline
d\'ecroissante?&&&&\\ \hline
scroissante?&&&&\\ \hline
sd\'ecroissante?&&&&\\ \hline
semboit\'ee?&&&&\\ \hline
sd\'eboit\'ee?&&&&\\ \hline
intersection&&&&\\ \hline
r\'eunion&&&&\\ \hline
\end {tabular}


	\end{document}


~	


\begin{tabular}{|l|c|c|c|c|}
\hline
&u_0=$1000$&$-1000$&$2$&$1,5$\\ \hline
injective?&~~~~~~~~~~~~~~~~~~~~~~~~~&~~~~~~~~~~~~~~~~~~~~~~~~~~
&~~~~~~~~~~~~~~~~~~~~~~~~~~&~~~~~~~~~~~~~~~~~~~~~~~~~~\\ \hline
image&&&&\\ \hline
$\#f_i^{-1}(3)$&&&&\\ \hline
major\'ee?&&&&\\ \hline
minor\'ee?&&&&\\ \hline
born\'ee?&&&&\\ \hline
max&&&&\\ \hline
min&&&&\\ \hline
sup&&&&\\ \hline
inf&&&&\\ \hline
\end {tabular}

	\end{document}
