Let f (x) = LNX, G (x) = f (x) + F '(x). (I) find the monotone interval and minimum value of G (x); (II) discuss the size relationship between G (x) and G (LX); (III) find the value range of a, so that G (a) - G (x) < 1A holds for any x > 0 | Math enthusiast | Mathematical art

Let f (x) = LNX, G (x) = f (x) + F '(x). (I) find the monotone interval and minimum value of G (x); (II) discuss the size relationship between G (x) and G (LX); (III) find the value range of a, so that G (a) - G (x) < 1A holds for any x > 0

(I) Let f (x) = LNX, G (x) = LNX + 1 x, G '(x) = X-1 x 2, let g' (x) = 0 get x = 1, when x ∈ (0,1), G '(x) < 0, so (0,1) is the monotone decreasing interval of G (x). When x ∈ (1, + ∞), G' (x) > 0, so (1, + ∞) is the monotone recurrence interval of G (x)

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