The four options all consider the same problem, that is, the mean inequality, a 2+b 2 ≥ 2ab, or it can be written as a+b≥2√(ab), where both a and b are positive numbers, and the condition of equal sign is a = b.

Option A is wrong, because lgx may be positive or negative.

When x > 1, lgx is positive, lgx+ 1/lgx≥2, and when x= 10, the equal sign holds.

When 0 < x < 1, lgx is negative, so -lgx- 1/lgx≥2, and when x =110, the equal sign holds, so lgx+ 1/lgx≤-2.

Option b, correct, √x is a positive number, so √ x+1√ x ≥ 2, and the equal sign holds when x= 1.

Option c, error. X+ 1/x≥2, and the equal sign holds when x= 1. Here x≥2, so x+ 1/x > 2, 2 is not the minimum value of x+ 1/x.

Option d, according to the definition, it can be judged that x- 1/x is increasing function. When x=2, x- 1/x gets the maximum value of 3/2.