Let f (x) = X3 + bx2 + CX (x ∈ R), if G (x) = f (x) - F ′ (x) is an odd function, (1) find the value of B and C; (2) find the monotone interval of G (x)

Let f (x) = X3 + bx2 + CX (x ∈ R), if G (x) = f (x) - F ′ (x) is an odd function, (1) find the value of B and C; (2) find the monotone interval of G (x)

(1) ∵ f (x) = X3 + bx2 + CX, ∵ f '(x) = 3x2 + 2bx + C. Thus g (x) = f (x) - f' (x) = X3 + bx2 + CX - (3x2 + 2bx + C) = X3 + (B-3) x2 + (c-2b) x-C is an odd function, so g (0) = 0 is C = 0, and B = 3 is defined by the odd function; (2) g (x) = x3-6x is known from (1), so G '(x) = 3x2-6, when G' (x) > 0, x ＜ - 2 or X ＞ 2, when G '(x) ＜ 0, - 2 ＜ x ＜ 2 From this, we can see that (- ∞, - 2) and (2, + ∞) are monotone increasing intervals of function g (x); (- 2,2) are monotone decreasing intervals of function g (x);

The function f (x) = the cube of X + the square of BX + CX is monotonically decreasing on [- 1,2]. It is proved that B + C is greater than or equal to - 15 / 2
(1 / 2) the following three equations are known: square of X + 4ax + 3 = 0, square of X + (A-1) square of X + a = 0, square of X + 2ax-a = 0. If there is at least one real root, the value range of real number a (2 / 2) can be obtained

Proof: the derivative of F (x) = the square of 3x + 2bx + C
Because it decreases monotonically on [- 1,2]
So: 3 * (- 1) * (- 1) + 2B * (- 1) + C

Let f (x) and G (x) satisfy f (- x) = - f (x), G (- x) = g (x) respectively. If f (x) + G (x) = x * 2 + X + 1, find f (x)
Given that f (x) and G (x) satisfy f (- x) = - f (x), G (- x) = g (x) respectively, if f (x) + G (x) = x * 2 + X + 1, find the analytic expressions of F (x) and G (x)
Let a = {x | x ^ 2 + (a + 2) x + 1 = 0, X belong to R}. If the intersection of a and negative real number is equal to an empty set, then the value range of A
A = {x | x ≥ 1 + A * 2 or X ≤ 1-A * 2}, B = {x | half

1.①.f(x)+g(x)=x^2+x+1
②.f(-x)+g(-x)=(-x)^2-x+1
That is, ③. - f (x) + G (x) = x ^ 2-x + 1
①+③:
2g(x)=2*x^2+2
g(x)=x^2+1
So f (x) = X
2. If △ 0, a is an empty set, so the intersection set is an empty set
Jiezhide - 40
so a