It is known that the function y = f (x) is an odd function on R and an increasing function on (0, + ∞). It is proved that: (1) f (0) = 0; (2) y = f (x) is also an increasing function on (- ∞, 0)

It is proved that: (1) f (x) & nbsp; If x 1 ＜ x 2 ＜ 0, then - x 1 ＞ - x 2 ＞ 0, ∵ f (x) is an increasing function on (0, + ∞), f (- x 1) ＞ f (- x 2), and f (x) is an odd function on R, then - f (x 1) ＞ - f (x 2), that is, f (x 1) ＜ f (x 2) and y = f (x) are increasing functions on (- ∞, 0) Count

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1. Find the set of independent variables X that make the function y = 3cos (2x + π / 4) obtain the maximum and minimum values, and write out the maximum and minimum values respectively
2. Find the maximum and minimum of function f (x) = √ 3cos & sup2; X + sinxcosx
3. For all real numbers x and y, if the function y = f (x), satisfies f (XY) = f (x) f (y), and f (0) is not equal to 0, find f (2009) = () Others say: f(0) = f(0)*f(0) => f(0) = 1 f(0) = f(0) * f(2009) = f(2009) = 1 But shouldn't 2009 and 0 satisfy the relation between X and y?
4. If the plane vector a = (3,4), B = (sin α, cos α), and a ‖ B, then cos2x=
5. Let f (x) = vector a · B, where vector a = (2, cos2x) and B = (1, - 2) find the monotone interval of F (x). When x belongs to [0, π / 3], find the range of F (x). To ensure the accuracy of the method, the following formula is used
6. If f (x) satisfies f (a × b) = f (a) + F (b) (a, B belongs to R) and f (2) = m, f (3) = n, then f (72)=
7. If m, n ∈ [- 1, 1], M + n ≠ 0, then f (m) + F (n) m + n & gt; 0. (1) judge the monotonicity of F (x) on [- 1, 1], and prove your conclusion; (2) solve the inequality: F (x + 12) & lt; F (1 x − 1); (3) if f (x) ≤ t 2-2 at + 1 holds for all x ∈ [- 1,1], a ∈ [- 1,1], find the value range of real number t
8. The function f (x) defined on (- 1,1) satisfies: for any x, y belonging to (- 1,1), there is f (x) + F (y) = f (x + Y1 + XY). (1) proof: the function f (x) is an odd function! (2) If x belongs to (- 1,0), f (x) > 0. Prove that f (x) is a decreasing function on (- 1,1)
9. Given that f (x) = ax / 2x + 3, (x is not equal to two-thirds of negative), satisfy f (f (x)) = x, find the value of A. (the outside is bracket, the inside is bracket)
10. Given the function f (x) = LG [(A2-1) x2 + (a + 1) x + 1] (1) if the domain of F (x) is r, find the value range of real number a; (2) if the domain of F (x) is r, find the value range of real number a
11. F (x) is an odd function and an increasing function in the interval (0, positive infinity), and f (- 2) = 0, f (x-1)
12. It is proved that the function f (x) = - x 3 + 1 is a decreasing function on (- ∞, + ∞)
13. It is proved that the function f (x) = - x ^ 3 + 1 is a monotone decreasing function on R
14. It is known that the function f (x) is a decreasing function in the domain of definition (- infinity, 4) and can make f (m-sinx) ≤ f (1 + 2m) - 7 / 4 under the root sign)
15. Let f (x) = - 2x ^ 2 + 7X-2, for real number m (0 < m < 7 / 4), if the domain of definition and value of F (x) are [M, 3] and [1,3 / M] respectively, what is the value of M?
16. It is known that the range of the function f (x) = 1 / 2x ^ 2-x + 3 / 2 defined on [1, M] is also [1, M], then the value of the real number m is
17. The function y = f (x) defined on R, when x ＞ 0, f (x) ＞ 1, and for any a, B belongs to R, f (a + b) = f (a) f (b), (1) Find f (0) = 1; (2) Proof: for any x belongs to R, f (x) > 0 (3) It is proved that f (x) is an increasing function on R; (4) If f (x) · f (2x-x2) > 1, find the value range of X
18. The increasing function y = f (x) defined on R has f (x + y) = f (x) + F (y) for any x, y belonging to R Find f (0) To prove that f (x) is an odd function Solving inequality f (3x) + F (x + 1) < 0
19. The function defined on R is y = f (x), and f (0) is not equal to 0. When x > 0, f (x) > 1, and for any a and B, f (0) = 1 is proved by F (a + b) = f (a) * f (b) (1) The function defined on R is y = f (x), f (0) is not equal to 0, when x > 0, f (x) > 1, and for any a, B, f (a + b) = f (a) * f (b) (1) Prove that f (0) = 1 (2) prove that for any x belongs to R, f (x) > 0 (3) prove that f (x) is an increasing function on R. 4) if f (x) * f (2x-x2) > 1, find the value range of X
20. The function defined on R is y = f (x), f (0) is not equal to 0, when x > 0, f (x) > 1, and for any a, B belongs to R, f (a + b) = f (a) f (b) (1) , prove that f (0) = 1; (2), prove that for any x belonging to R, f (x) > 0; (3), prove that f (x) is an increasing function on R; (4), if f (x) * f (2x-x Square) > 1, find X