The function f (x) = 1 / 3x3-x + C defined on R If the minimum value of the domain y = f (x) in the interval [0, positive infinity] is 7 / 3, find C

The function f (x) = 1 / 3x3-x + C defined on R If the minimum value of the domain y = f (x) in the interval [0, positive infinity] is 7 / 3, find C

f'(x)=x²-1
Let f '(x) > 0, then x > 1 or X

Let f (x) = 1 / 3x3-ax2 + (A2-1) x + B (a, B belong to R), if y = f (x)
Given the function f (x) = 1 / 3x3-ax2 + (A2-1) x + B (a, B belong to R), if the tangent equation of the image of y = f (x) at point (1, f (1)) is x + Y-3 = 0, find the maximum value in the interval [- 2,4]

f'(x)=x^2-2ax+a^2-1
The tangent slope is - 1
1-2a+a^2-1=-1
a=1
Tangent point x = 1, y = 2
1/3-1+b=2
b=8/3
f(x)=x^3/3-x^2+8/3
f'(x)=x(x-2)
x

Given the function f (x) = (9 ^ x + k * 3 ^ x + 1) / (9 ^ x + 3 ^ x + 1), when k = 1, for any real number, f (x1) = f (x2) = f (x3) = 1, then there are triangles with three sides of F (x1), f (x2), f (x3). When k > 1, if there are triangles with three sides of F (x1), f (x2), f (x3) for any real number, then the maximum value of real number k is____

4.
When k > 1, f (x) = (9 ^ x + k * 3 ^ x + 1) / (9 ^ x + 3 ^ x + 1) = 1 + (k-1) / (3 ^ x + 3 ^ (- x) + 1), so the range of F (x) is (1,1 + (k-1) / 3]. For any real number, there are triangles with three sides of F (x1), f (x2), f (x3), that is, f (x1) - f (x2) < f (x3) is constant. The maximum value of F (x1) - f (x2) is less than (k-1) / 3, and (k-1) / 3 ≤ 1, so K ≤ 4