(1) Substitute the original equation ALNX+BX =-2x- 1 and the tangent equation a/x+b=-2 into x= 1.

A = 1 and b =-3 can be found, and the function can be found to decrease in the range of x∈( 1/3, +∞) by using the tangent equation.

So f (1/3) is the largest, so f( 1/3)

(2) When y = 0, we can get lnx =-x 2+3x-2, and the symmetry axis of the equation on the right is x=2/3 and y= 1/4.

The ratio of ln3/2 to 1/4 (let LN2/3 = X. Then use x to 1/4. Because E X is increasing function, we can compare (2/3) 4 with E X to E (1/4).

So there is only one intersection (1, 0), so the maximum value of k is 1.