f'(x) = 3x² - 2ax + 3 = 0
在[1,+∞)上是增函数,有两种可能:
(1) 3x² - 2ax + 3 恒≥ 0
∆ = 4(a² - 9) ≤ 0,-3 ≤ a ≤ 3
(2)
3x² - 2ax + 3 = 0的较大根x₂ = [a + √(4a² - 36)]/6 = [a + √(a² - 9)]/3 ≤ 1
√(a² - 9) ≤ 3 - a
显然a > 3时,不等式不成立
a < 3:a² - 9 ≤ a² - 6a + 9
a ≤ 3
结合(1)(2):a ≤ 3