It is known that the function f (x) = X3 + AX2 + BX + 4 is an increasing function on (- ∞, 0) and a decreasing function on (0, 1). (I) find the value of B; (II) when x ≥ 0, the curve y = f (x) is always above the straight line y = a2x-4, and find the value range of A

It is known that the function f (x) = X3 + AX2 + BX + 4 is an increasing function on (- ∞, 0) and a decreasing function on (0, 1). (I) find the value of B; (II) when x ≥ 0, the curve y = f (x) is always above the straight line y = a2x-4, and find the value range of A

(I) ∵ f (x) = X3 + AX2 + BX + 4, ∵ f ′ (x) = 3x2 + 2aX + B. ∵ f (x) is an increasing function on (- ∞, 0) and a decreasing function on (0, 1), ∵ when x = 0, f (x) has a maximum, that is, f ′ (x) = 0, ∵ B = 0. (II) f ′ (x) = 3x2 + 2aX = x (3x + 2a), ∵ f (x) is an increasing function on (- ∞, 0), a decreasing function on (0, 1), ∵ 23a ≥ 1, that is, a Let g (x) = (X3 + AX2 + 4) - (a2x-4), where x ∈ [0, + ∝, G (x) ≥ 0 holds. Let g ′ (x) = 3x2 + 2ax-a2 = (3x-a) (x + a), let g ′ (x) = 0, two roots - A, A3, and A3 < 0 ＜ a, X (0, - a) - A (- A, + ∞) g ′ (x) - 0 + G (x) Let g (- a) = (- A3 + a3 + 4) - (- a3-4) > 0, A3 ＞ - 8, and let a ≤ − 32, 2 < a ≤ & nbsp; − 32