Réponse :
Explications étape par étape
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Déterminer le tableau de variation des fonctions suivantes
F(x):x3 - 4x(au carré) + 4x +2
F(x) = x^3 - 4x^2 + 4x + 2
F’(x) = 3x^2 - 8x + 4
[tex]\Delta = (-8)^{2} - 4 * 3 * 4 = 64 - 48 = 16[/tex]
X1 = (8 - 4)/(2 * 3) = 4/6 = 2/3
X2 = (8 + 4)/6 = 12/6 = 2
X...........|-inf..............2/3..............2............+inf
F’(x).....|..........(+)........o......(-).......o......(+).......
F(x)......|////////////F(2/3)\\\\\\\\\F(2)////////////
/// : croissante
\\\ : decroissante
F(2/3) = (2/3)^3 - 4(2/3)^2 + 4(2/3) + 2
F(2/3) = 8/27 - 4 * 4/9 + 8/3 + 2
F(2/3) = 8/27 - 16/9 + 72/27 + 54/27
F(2/3) = 8/27 - 48/27 + 126/27
F(2/3) = 134/27 - 48/27
F(2/3) = 86/27
F(2) = (2)^3 - 4(2)^2 + 4(2) + 2
F(2) = 8 - 4 * 4 + 8 + 2
F(2) = 8 - 16 + 10
F(2) = 18 - 16
F(2) = 2
G(x):(7x(au carré)+8x-3)(8-7x)
G(x) = (7x^2 + 8x - 3)(8 - 7x)
G(x) = 56x^2 - 49x^3 + 64x - 56x^2 - 24 + 21x
G(x) = -49x^3 + 85x - 24
G’(x) = -147x^2 + 85
[tex]\Delta = 0 - 4 * (-147) * 85 = 2940[/tex]
[tex]\sqrt{\Delta} \approx 54,22[/tex]
X1 = (-54,22)/(2 * -147) = 0,18
X2 = (54,22)/(-294) = -0,18
X............|-inf............(-0,18)...........0,18......+inf
G’(x).......|.........(-)........o.......(+).......o...(-)..........
G(x).......|\\\\\\\\\\\G(-0,18)//////G(0,18)\\\\\\\\
G(-0,18) = -49(-0,18)^3 + 85(-0,18) - 24
G(-0,18) = 0,29 - 15,3 - 24
G(-0,18) = (-39,01)
G(0,18) = -49(0,18)^3 + 85(0,18) - 24
G(0,18) = -0,29 + 15,3 - 24
G(0,18) = (-8,99)