Given that the function f (x) = (AX2 + x) - xlnx increases monotonically on [1, + ∞), then the value range of real number a is______ ．

Given that the function f (x) = (AX2 + x) - xlnx increases monotonically on [1, + ∞), then the value range of real number a is______ ．

If f '(x) = 2aX LNX ∵ function f (x) = (AX2 + x) - xlnx is monotonically increasing on [1, + ∞), f' (x) = 2aX LNX ≥ 0 is constant on [1, + ∞), let g (x) = lnxx (x ＞ 0), then G '(x) = 1 − lnxx2, let g' (x) ＞ 0, then 0 ＜ x ＜ E; let g '(x) ＜ 0, then x ＞ E; let g' (E) monotonically increase on (0, e), then G '(x) ＞ E+ When ∧ x = e, the maximum value of the function is 1E ∧ 2A ≥ 1E ∧ a ≥ 12e, so the answer is: [12e, + ∞)