Решение
6.
а) sin3x = √3/2
3x = (-1)^n arcsin(√3/2) + πn, n ∈ Z
3x = (-1)^n (π/3) + πn, n ∈ Z
x = (-1)^n (π/9) + π/3, n ∈ Z
б) sin²x - sinx = 0
sinx(sinx - 1) = 0
sinx = 0 sinx - 1 = 0
x₁= πk, k ∈Z sinx = 1
x₂ = π/2 + 2πn, n ∈ Z
7) lg(5x - 4) ≥ lg(3x - 7)
ОДЗ:
5x - 4 > 0, x > 0,8
3x - 7 > 0, x > 2(1/3)
x ∈ (2(1/3); + ∞)
10 > 0
5x - 4 ≥ 3x - 7
5x - 3x ≥ - 7 + 4
2x ≥ - 3
x ≥ - 1,5
Ответ: x ∈ (2(1/3); + ∞)
8) y = √(27 - x³)
27 - x³ ≥ 0
x³ ≤ 27
x ≤ 3
x∈ (- ∞; 3]
9) f(x) = 3x⁴ + 2x - 2√x - 1
f`(x) = 12x³ + 2 - 2/2√x = 12x³ + 2 - 1/√x
f`(1) = 12 + 2 - 1 = 13