f (x) = 3 x² - 6 x + 7
soit x et x' ∈ R tel que x < x'
a) factoriser f(x') - f(x)
f(x') - f(x) = 3 x'² - 6 x' + 7 - (3 x² - 6 x + 7)
= 3 x'² - 3 x² - 6 x' + 6 x
= 3(x'² - x²) - 6(x' - x)
= 3 (x' + x)(x' - x) - 2(x' - x)
= (x' - x)(3(x' + x) - 6)
= (x' - x)(3 x' + 3 x - 6)
= 3(x' - x)(x' + x - 2)
b) Tableau de variation de f sur R
f '(x) = 6 x - 6 ⇒ f '(x) = 0 = 6(x - 1) ⇒ x = 1
f(1) = 3(1) - 6(1) + 7 = 4
x - ∞ 1 + ∞
f(x) +∞→→→→→→ 4→→→→→→ +∞
décroissante croissante
2) a) x ∈[- 2 ; 0] ⇔ -2 ≤ x ≤ 0
b) f(0) = 7 f(- 2) = 3(- 2)² - 6(-2) + 7 = 31
quand x ∈ [- 2 ; 0] ⇒ 7 ≤ f(x) ≤ 31
