f'(x)=(ax？ -2x+ 1)/x

Molecular δ = 4-4a

When δ≤ 0 →a≥ 1, f'(x)≥0 f(x) is the increasing function of the monotonically increasing interval x∈(0, +∞) (blue).

When δ > 0, that is, A.

0<a< When 1, define two stagnation points x? =[ 1-√( 1-a)]/a, left+right-,this is the maximum point.

x？ =[ 1-√( 1-a)]/a, left and right+,this is the minimum point.

Monotone increasing interval x∈(0, x? )∩(x？ +∞), monotonically decreasing interval x∈(x? ，x？ (red)

When a≤0, the stationary point X in the domain is defined. =[ 1-√( 1-a)]/a (or x? =? , when a=0), left+right-is the maximum point.

Monotone increasing interval x∈(0, x? ), monotonically decreasing interval x∈(x? , +∞) (black)