Let f (x) satisfy f (x + x ^ - 1) = x ^ 3 + x ^ - 3, then the expression of F (x) is

Let f (x) satisfy f (x + x ^ - 1) = x ^ 3 + x ^ - 3, then the expression of F (x) is

According to a ^ 3 + B ^ 3 = (a + b) (a ^ 2-AB + B ^ 2)
（x+x^-1)^2=x^2+x^(-2) +2
So you should know

Given the function f (x) = the third power of X + 1, find the expression of F (f (x) - 1)

f(x)=x^3+1
Then f (x) - 1 = x ^ 3
f(f(x)-1)=f(x^3)=(x^3)^3+1=x^9+1

It is known that the derivative of cubic function f (x) is f '(x), and f' (1) = 0, f '(2) = 3, f' (3) = 12. (I) find the expression of F (x) - f (0); (II) if f (x) > F '(x) holds for any x ∈ [- 1,4], find the value range of F (0)

(I) Let f (x) = AX3 + bx2 + CX + D, then f ′ (x) = 3ax2 + 2bx + c.. 3A + 2B + C = 012a + 4B + C = 327a + 6B + C = 12, i.e. a = 1b = − 3C = 3.. f (x) - f (0) = x3-3x2 + 3x. (II) f ′ (x) = 3x2-6x + 3, ∵ for any x ∈ [- 1,4], f (x) > F ′ (x) holds