Given the function f (x) = ln (1 + x) - X1 + X. (1) find the minimum of F (x); (2) if a, b > 0, prove: LNA LNB ≥ 1-ba

Given the function f (x) = ln (1 + x) - X1 + X. (1) find the minimum of F (x); (2) if a, b > 0, prove: LNA LNB ≥ 1-ba

(1) F ′ (x) = 11 + X-1 (1 + x) 2 = x (1 + x) 2, X ＞ - 1. When - 1 < x < 0, f ′ (x) < 0, f (x) monotonically decreases on (- 1, 0); when x = 0, f ′ (x) = 0; when x > 1, f ′ (x) > 0, f (x) monotonically increases on (1, + ∞), so x = 1 is the minimum of F (x), so the minimum of F (x) = f (0) = 0; (2) from (1), f (x) is the minimum of F (x)（ x) Let ln (1 + x) ≥ X1 + X be tenable in the domain (- 1, + ∞). Let LNA LNB ≥ 1-ba be tenable, that is, let lnab ≥ 1-ba be tenable. Let 1 + x = AB, then X1 + x = 1-1x + 1 = 1-ba, then lnab ≥ 1-ba, the inequality is tenable