Ainsi, la condition (ssi) pour que R_n(f) soit croissante pour un polynome de degré 3 peuvent se réécrire comme
47
Avaldis 48: "f" left parenthesis, 1 , right parenthesis less than min left parenthesis, "f" left parenthesis, 0 , right parenthesis , "f" left parenthesis, 1 half , right parenthesis , right parenthesisf1<minf0,f12
48
Et la sym est concave croissante ssi
49
Avaldis 50: "f" left parenthesis, 1 , right parenthesis less than or equal to 2 "f" left parenthesis, 1 half , right parenthesisf1≤2f12
50
Avaldis 51:
51
Avaldis 52:
52
Avaldis 53: StartFraction, left parenthesis, "n" plus 1 , right parenthesis left parenthesis, 2 "a" left parenthesis, 6 "n" cubed plus 9 "n" squared plus "n" minus 1 , right parenthesis plus 5 "n" squared left parenthesis, 3 "b" left parenthesis, "n" plus 1 , right parenthesis plus 2 left parenthesis, 2 "c" "n" plus "c" plus 3 "d" "n" , right parenthesis , right parenthesis , right parenthesis Over 60 "n" to the 4th power , EndFractionn+12a6n3+9n2+n−1+5n23bn+1+22cn+c+3dn60n4
53
Avaldis 54: StartFraction, "n" left parenthesis, 2 "a" left parenthesis, 6 left parenthesis, "n" minus 1 , right parenthesis cubed plus 9 left parenthesis, "n" minus 1 , right parenthesis squared plus "n" minus 2 , right parenthesis plus 5 left parenthesis, "n" minus 1 , right parenthesis squared left parenthesis, 3 "b" "n" plus 2 left parenthesis, 2 "c" "n" minus "c" plus 3 "d" left parenthesis, "n" minus 1 , right parenthesis , right parenthesis , right parenthesis , right parenthesis Over 60 left parenthesis, "n" minus 1 , right parenthesis to the 4th power , EndFractionn2a6n−13+9n−12+n−2+5n−123bn+22cn−c+3dn−160n−14
54
Avaldis 55: StartFraction, left parenthesis, "n" plus 1 , right parenthesis left parenthesis, 2 "a" left parenthesis, 6 "n" cubed plus 9 "n" squared plus "n" minus 1 , right parenthesis plus 5 "n" squared left parenthesis, 3 "b" left parenthesis, "n" plus 1 , right parenthesis plus 2 left parenthesis, 2 "c" "n" plus "c" plus 3 "d" "n" , right parenthesis , right parenthesis , right parenthesis Over 60 "n" to the 4th power , EndFraction minus StartFraction, "n" left parenthesis, 2 "a" left parenthesis, 6 left parenthesis, "n" minus 1 , right parenthesis cubed plus 9 left parenthesis, "n" minus 1 , right parenthesis squared plus "n" minus 2 , right parenthesis plus 5 left parenthesis, "n" minus 1 , right parenthesis squared left parenthesis, 3 "b" "n" plus 2 left parenthesis, 2 "c" "n" minus "c" plus 3 "d" left parenthesis, "n" minus 1 , right parenthesis , right parenthesis , right parenthesis , right parenthesis Over 60 left parenthesis, "n" minus 1 , right parenthesis to the 4th power , EndFractionn+12a6n3+9n2+n−1+5n23bn+1+22cn+c+3dn60n4−n2a6n−13+9n−12+n−2+5n−123bn+22cn−c+3dn−160n−14
55
Avaldis 56: StartFraction, negative 30 left parenthesis, "a" plus "b" plus "c" plus "d" , right parenthesis "n" to the 6th power plus 10 left parenthesis, 5 "a" plus 6 "b" plus 7 "c" plus 9 "d" , right parenthesis "n" to the 5th power plus 5 left parenthesis, 2 "a" minus 3 "b" minus 8 "c" minus 18 "d" , right parenthesis "n" to the 4th power plus 2 left parenthesis, 15 "d" minus 15 "b" minus 5 "c" minus 21 "a" , right parenthesis "n" cubed plus left parenthesis, 8 "a" plus 15 "b" plus 10 "c" , right parenthesis "n" squared plus 8 "a" "n" minus 2 "a" Over 60 "n" to the 4th power left parenthesis, "n" minus 1 , right parenthesis to the 4th power , EndFraction−30a+b+c+dn6+105a+6b+7c+9dn5+52a−3b−8c−18dn4+215d−15b−5c−21an3+8a+15b+10cn2+8an−2a60n4n−14
56
Avaldis 57: StartFraction, negative 30 left parenthesis, "a" plus "b" plus "c" plus "d" , right parenthesis "x" to the 6th power plus 10 left parenthesis, 5 "a" plus 6 "b" plus 7 "c" plus 9 "d" , right parenthesis "x" to the 5th power plus 5 left parenthesis, 2 "a" minus 3 "b" minus 8 "c" minus 18 "d" , right parenthesis "x" to the 4th power plus 2 left parenthesis, 15 "d" minus 15 "b" minus 5 "c" minus 21 "a" , right parenthesis "x" cubed plus left parenthesis, 8 "a" plus 15 "b" plus 10 "c" , right parenthesis "x" squared plus 8 "a" "x" minus 2 "a" Over 60 "x" to the 4th power left parenthesis, "x" minus 1 , right parenthesis to the 4th power , EndFraction−30a+b+c+dx6+105a+6b+7c+9dx5+52a−3b−8c−18dx4+215d−15b−5c−21ax3+8a+15b+10cx2+8ax−2a60x4x−14
57
Avaldis 58: "a" plus "b" plus "c" plus "d" less than 0a+b+c+d<0
58
Avaldis 59:
59
Avaldis 60: "T" left parenthesis, "x" , right parenthesis equals StartFraction, left parenthesis, "x" plus 1 , right parenthesis left parenthesis, 2 "a" left parenthesis, 6 "x" cubed plus 9 "x" squared plus "x" minus 1 , right parenthesis plus 5 "x" squared left parenthesis, 3 "b" left parenthesis, "x" plus 1 , right parenthesis plus 2 left parenthesis, 2 "c" "x" plus "c" plus 3 "d" "x" , right parenthesis , right parenthesis , right parenthesis Over 60 "x" to the 4th power , EndFractionTx=x+12a6x3+9x2+x−1+5x23bx+1+22cx+c+3dx60x4
60
Avaldis 61: "T" prime left parenthesis, "x" , right parenthesisT′x
61
Avaldis 62: negative 2 StartFraction, 15 left parenthesis, "a" plus "b" plus "c" plus "d" , right parenthesis "x" cubed plus 5 left parenthesis, 4 "a" plus 3 "b" plus 2 "c" , right parenthesis "x" squared minus 4 "a" Over 60 "x" to the 5th power , EndFraction−215a+b+c+dx3+54a+3b+2cx2−4a60x5
62
Avaldis 63: "a" plus "b" plus "c" plus "d" less than 0a+b+c+d<0
63
on veut que
64
Avaldis 65: 15 left parenthesis, "a" plus "b" plus "c" plus "d" , right parenthesis "x" cubed plus 5 left parenthesis, 4 "a" plus 3 "b" plus 2 "c" , right parenthesis "x" squared minus 4 "a"15a+b+c+dx3+54a+3b+2cx2−4a
65
soit négatif.
66
Si 0>17a+15b+13c+9d, alors la fonction est croissante jusqu'à décroissante
67
Avaldis 68: "x" equals negative StartFraction, 2 left parenthesis, 4 "a" plus 3 "b" plus 2 "c" , right parenthesis Over 9 left parenthesis, "a" plus "b" plus "c" plus "d" , right parenthesis , EndFractionx=−24a+3b+2c9a+b+c+d
68
qui est plus petit que 1 et décroissante après. Ainsi, elle est décroissante pour tout n>1 et il suffit de vérifier que
69
Avaldis 70: 31 "a" plus 30 "b" plus 25 "c" plus 15 "d" less than or equal to 031a+30b+25c+15d≤0
70
Si 0<17a+15b+13c+9d, alors la fct est croissante jusqu'à
71
Avaldis 72: negative StartFraction, 2 left parenthesis, 4 "a" plus 3 "b" plus 2 "c" , right parenthesis Over 9 left parenthesis, "a" plus "b" plus "c" plus "d" , right parenthesis , EndFraction greater than 1−24a+3b+2c9a+b+c+d>1
72
et décroissante après. Dans ce cas, il suffit de vérifier que l'entier inférieur et supérieur au maximum est négatif. Mais par soucis de simplicité, on peut également seulement vérifier le point exact même si ce n'est pas un entier. On obtient alors
73
Avaldis 74: StartFraction, 20 left parenthesis, 4 "a" plus 3 "b" plus 2 "c" , right parenthesis cubed Over 243 left parenthesis, "a" plus "b" plus "c" plus "d" , right parenthesis squared , EndFraction minus 4 "a" less than 0204a+3b+2c3243a+b+c+d2−4a<0
74
Avaldis 75:
75
En résumé, on veut soit
76
Avaldis 77: "a" plus "b" plus "c" plus "d" less than or equal to 0 & 17 "a" plus 15 "b" plus 13 "c" plus 9 "d" less than or equal to 0 & 31 "a" plus 30 "b" plus 25 "c" plus 15 "d" less than or equal to 0a+b+c+d≤0&17a+15b+13c+9d≤0&31a+30b+25c+15d≤0
77
ou soit
78
Avaldis 79: "a" plus "b" plus "c" plus "d" less than or equal to 0 & 17 "a" plus 15 "b" plus 13 "c" plus 9 "d" greater than or equal to 0 & StartFraction, 20 left parenthesis, 4 "a" plus 3 "b" plus 2 "c" , right parenthesis cubed Over 243 left parenthesis, "a" plus "b" plus "c" plus "d" , right parenthesis squared , EndFraction minus 4 "a" less than or equal to 0a+b+c+d≤0&17a+15b+13c+9d≥0&204a+3b+2c3243a+b+c+d2−4a≤0
79
Avaldis 80: 17 "a" plus 15 "b" plus 13 "c" plus 9 "d"17a+15b+13c+9d
80
Avaldis 81: 17 "f" left parenthesis, 1 , right parenthesis minus 16 "f" left parenthesis, 1 half , right parenthesis17f1−16f12
81
Avaldis 82: "a" plus "b" plus "c" plus "d" less than min left parenthesis, 0 , StartFraction, "a" plus 2 "b" plus 4 "c" plus 8 "d" Over 16 , EndFraction , right parenthesisa+b+c+d<min0,a+2b+4c+8d16
82
Avaldis 83: 15 "a" plus 14 "b" plus 12 "c" plus 8 "d" less than 015a+14b+12c+8d<0
83
Avaldis 84: 4 "a" plus 3 "b" plus 2 "c" less than 04a+3b+2c<0
84
Avaldis 85:
85
Avaldis 86: "g" left parenthesis, "x" , right parenthesis equals StartFraction, 1 Over 1 minus "b" Subscript, 0 , Baseline "x" plus "x" squared , EndFractiongx=11−b0x+x2
Avaldis 88: "g" left parenthesis, 1 , right parenthesisg1
88
Avaldis 89: min left parenthesis, "g" left parenthesis, 0 , right parenthesis , "g" left parenthesis, 1 half , right parenthesis , right parenthesisming0,g12
89
Avaldis 90: StartFraction, 1 Over 2 minus "x" , EndFraction less than or equal to min left parenthesis, 1 , StartFraction, 4 Over 5 minus 2 "x" , EndFraction , right parenthesis12−x≤min1,45−2x
