The function f (x) = alnx-bx ^ 2 (x ≥ 0) when B = 0, if the inequality f (x) ≥ m + X holds for all a ∈ [0,3 / 2], X ∈ (1, e ^ 2), find the range of M

The function f (x) = alnx-bx ^ 2 (x ≥ 0) when B = 0, if the inequality f (x) ≥ m + X holds for all a ∈ [0,3 / 2], X ∈ (1, e ^ 2), find the range of M

When B = 0, f (x) = alnx, if the inequality f (x) ≥ m + X holds for all a ∈ [0,3 / 2], X ∈ (1, e ^ 2],
Then alnx ≥ m + X holds for all a ∈ [0,3 / 2], X ∈ (1, e ^ 2),
That is, m ≤ alnx-x holds for all a ∈ [0,3 / 2], X ∈ (1, e ^ 2),
Let H (a) = alnx-x = (LNX) A-X, then H (a) is a function of degree a,
∵x∈（1,e^2],
∴lnx＞0
H (a) monotonically increasing
∵a∈［0,3/2］,
∴h（a）min=h（0）=-x
For all x ∈ (1, E2], m ≤ - x holds,
∵1＜x＜e^2,
∴-e^2≤-x＜-1,
∴m≤（-x）min=-e^2
m≦-e²