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" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 
4 "" 0 "" {TEXT -1 22 "Nombre de solution de " }{XPPEDIT 18 0 "sin^2;
" "6#*$)%$sinG\"\"#\"\"\"" }{TEXT -1 13 "(x) +cos(x)=m" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Si m > " }{XPPEDIT 
18 0 "5/4;" "6#*&\"\"&\"\"\"\"\"%!\"\"" }{TEXT -1 18 ", pas de solutio
ns" }}{PARA 0 "" 0 "" {TEXT -1 11 "si 1 < m < " }{XPPEDIT 18 0 "5/4;" 
"6#*&\"\"&\"\"\"\"\"%!\"\"" }{TEXT -1 13 ", 4 solutions" }}{PARA 0 "" 
0 "" {TEXT -1 20 "si m=1,  3 solutions" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 4 "si  " }{XPPEDIT 18 0 "-1 <= m;" "6#1,$
\"\"\"!\"\"%\"mG" }{TEXT -1 17 "< 1 , 2 solutions" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "si m < -1 pas de solution
s" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 33 "
Exercice 4 : Inegelite de Huygens" }}{PARA 0 "" 0 "" {TEXT -1 28 "Prou
ver que pour toux de [0;" }{XPPEDIT 18 0 "Pi/2;" "6#*&%#PiG\"\"\"\"\"#
!\"\"" }{TEXT -1 3 "[, " }{XPPEDIT 18 0 "3*x <= 2*sin(x)+tan(x);" "6#1
*&\"\"$\"\"\"%\"xGF&,&*&\"\"#\"\"\"-%$sinG6#%\"xGF+F+-%$tanG6#F/F+" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "g:= x -> 3*x
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGR6#%\"xG6\"6$%)operatorG%&a
rrowGF(,$9$\"\"$F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "h
:= x -> 2*sin(x)+tan(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hGR6#%
\"xG6\"6$%)operatorG%&arrowGF(,&-%$sinG6#9$\"\"#-%$tanGF/\"\"\"F(F(F(
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "f:= x -> 2*sin(x)+sin(x
)/cos(x)-3*x;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"xG6\"6$%)
operatorG%&arrowGF(,(-%$sinG6#9$\"\"#*&F-\"\"\"-%$cosGF/!\"\"\"\"\"F0!
\"$F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 52 "Remarque : quel est l'ensemble de definition de f  ?" }}{SECT 
0 {PARA 4 "" 0 "" {TEXT -1 24 "Signe de la derivee de f" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "derivee
:=diff(f(x),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(deriveeG,(-%$cos
G6#%\"xG\"\"#!\"#\"\"\"*&*$)-%$sinGF(F*\"\"\"F2*$)F&\"\"#F2!\"\"F," }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "derivee:=derivee*cos(x)^2;
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(deriveeG*&,(-%$cosG6#%\"xG\"\"#
!\"#\"\"\"*&*$)-%$sinGF)F+\"\"\"F3*$)F'\"\"#F3!\"\"F-F-)F'F+F3" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "derivee:=simplify(derivee);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(deriveeG,(*$)-%$cosG6#%\"xG\"\"
$\"\"\"\"\"#*$)F(F.F-!\"$\"\"\"F2" }}}{PARA 0 "" 0 "" {TEXT -1 7 "Comm
e  " }{XPPEDIT 18 0 "cos^2;" "6#*$)%$cosG\"\"#\"\"\"" }{TEXT -1 85 "(x
) est toujours positif la derivee de f et 2*cos(x)^3-3*cos(x)^2+1 ont \+
le meme signe" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "On pose X=cos(x);" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "Dans le d
omaine d'etude X est compris entre 0 et 1." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 25 "Etude de P(X)=2X^3-3X^2
+1" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 17 "P:=2*X^3-3*X^2+1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%\"PG,(*$)%\"XG\"\"$\"\"\"\"\"#*$)F(F+F*!\"$\"\"\"F/" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "roots(P);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#7$7$#!\"\"\"\"#\"\"\"7$F(F'" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "Les racines de P sont -0.5 et 1
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "diff(P,X);" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#,&*$)%\"XG\"\"#\"\"\"\"\"'F&!\"'" }}}{PARA 0 "" 0 "
" {TEXT -1 33 "La derivee a pour racines 0 et 1." }}{PARA 0 "" 0 "" 
{TEXT -1 36 "Elle est donc negative entre 0 et 1." }}{PARA 0 "" 0 "" 
{TEXT -1 34 "P est donc decroissant sur [0;1]. " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "subs(X=0,P);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 12 "subs(X=1,P);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"
\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "S
ur [0;1] P est compris entre 0 et 1 et est donc positif." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 87 "De cette etude, on conclue que la derivee de f est posi
tive sur l'intervalle considere." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 41 "Donc f est croissante sur cet intervalle.
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "f(0)=0
." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "f es
t donc positive su l'intervalle d'etude" }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 3 "" 0 "" {TEXT -1 0 "" }}
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