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

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• 1. 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)=
• 2. 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
• 3. 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)
• 4. 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)
• 5. 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
• 6. 1. Let f (x) s be an odd function on R and satisfy f (x + 2) = - f (x). When 0 ≤ x ≤ 1, f (x) = x, then f (3.5) =? 2. If f (1 / x) = 1 / (x + 1), then f (x) =?
• 7. It is known that f (x) = x ^ 2 + X + 1 (1) Find the analytic expression of F (2x); (2) Find the analytic expression of F [f (2x)]; (3) For any x ∈ R, it is proved that f (- 1 / 2 + x) = f (- 1 / 2-x) always holds
• 8. If the function f (x) defined on R + satisfies f (x) + F (y) + 2XY (XY) = f (XY) / F (x + y) for any x, y ∈ R +, then f (2)=______
• 9. A problem of mathematical function in grade one of senior high school The known function f (x) = (- ax ^ 2 + ax + 1) / 2x, X ∈ [1, + ∞) (1) When a = - 2, find the minimum value of function f (x) (2) If f (x) > 0 holds for any x ∈ [1, + ∞), try to find the value range of real number a
• 10. Equation: two of X 2-A ax + 4 = 0 are greater than 1, find the value range of real number a
• 11. If the plane vector a = (3,4), B = (sin α, cos α), and a ‖ B, then cos2x=
• 12. 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?
• 13. Find the maximum and minimum of function f (x) = √ 3cos & sup2; X + sinxcosx
• 14. 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
• 15. 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)
• 16. F (x) is an odd function and an increasing function in the interval (0, positive infinity), and f (- 2) = 0, f (x-1)
• 17. It is proved that the function f (x) = - x 3 + 1 is a decreasing function on (- ∞, + ∞)
• 18. It is proved that the function f (x) = - x ^ 3 + 1 is a monotone decreasing function on R
• 19. 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)
• 20. 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?