Let f (x) = ln (x + 1) + ax, (a ∈ R and a ≠ 0). (I) discuss the monotonicity of function f (x); (II) if a = 1, it is proved that f (x) − 3 ＜ LX holds when x ∈ [1,2]

Let f (x) = ln (x + 1) + ax, (a ∈ R and a ≠ 0). (I) discuss the monotonicity of function f (x); (II) if a = 1, it is proved that f (x) − 3 ＜ LX holds when x ∈ [1,2]

(I) the definition domain of function f (x) is (- 1, + ∞), f ′ (x) = 1 x + 1 + A. when a > 0, f ′ (x) > 0, f (x) is an increasing function on (0, + ∞); when a < 0, f ′ (x) > 0 leads to − 1 ＜ x ＜− 1A; when f ′ (x) ＜ 0 leads to X ＞ − 1a, f (x) is an increasing function on (− 1, − 1a), and a decreasing function on (− 1a, + ∞); (II) when a = 1, f (x) ＜ 1 ）=Ln (x + 1) + X if x ∈ [1,2], f (x) − 3 ＜ LX holds, that is to say, ln (x + 1) + X-1 x-3 ＜ 0 holds on [1,2], let g (x) = ln (x + 1) + X-1 x-3, it is easy to get that the function g (x) monotonically increases ∵ g (1) = 0 when x ∈ [1,2], then f (x) − 3 ＜ LX holds when G (x) ≥ 0 ∵ x ∈ [1,2]

Given the function f (x) = ln (AX + 1) + x ^ 3-x ^ 2-ax, 3. If a = - 1, the equation f (1-x) - (1-x) ^ 3 = B / X has real roots, find
The known function f (x) = ln (AX + 1) + x ^ 3-x ^ 2-ax
1. If x = 2 / 3 is the extreme point of y = f (x), find the value of real number a
2. If y = f (x) is an increasing function in [1, + ∞), find the value range of real number a
3. If a = - 1, the equation f (1-x) - (1-x) ^ 3 = B / X has real roots, find the value range of real number B

Let g (x) = xlnx-x ^ 3 + x ^ 2 = B have a real root. Let g (x) = xlnx-x ^ 3 + x ^ 2, then G '(x) = - 3x ^ 2 + 2x + 1 + LNX notice that G' (1) = 0, then G '(x) = - 6x + 2 + 1 / x = (- 6x ^ 2 + 2x + 1) / X is easy to know that G' (x) first increases and then decreases from G '(x), and G' (x) tends to 0 when (1, + ∞) is a decreasing function and X tends to 0

Ln (x + 1 / a) - AX = 0 has two different sign roots

Let f (x) = ln (x + 1 / a) - ax, (&; 1 / A0), the function is an increasing function on (&; 1 / A, + ∞), then f (x) = 0 has at most one zero point, which is not satisfied with the problem, so it is excluded; ② when a > 0, ax + 1 > 0, Let f '(x) = 0, then x = 0, when x ∈ (- 1 / A, 0), f' (x) > 0, f (x) monotonically increases, when x ∈ (0, + ∞), f '(...)