It is known that f (x) = 2cos ^ 2x + 2 √ 3xinxcosx + a (x ∈ R) a is a constant (1) Finding the minimum positive period of F (x) (2) Finding monotone increasing interval of F (x) (3) If the sum of the maximum and minimum of F (x) is 3, find the value of A

It is known that f (x) = 2cos ^ 2x + 2 √ 3xinxcosx + a (x ∈ R) a is a constant (1) Finding the minimum positive period of F (x) (2) Finding monotone increasing interval of F (x) (3) If the sum of the maximum and minimum of F (x) is 3, find the value of A

1) 2cos ^ 2x + 2 √ 3sinxcosx + a = cos2x + 1 + √ 3sin2x + a = 2Sin (2x + π / 6) + 1 + A, the period is π
2）-π/2+2kπ

Mathematical function question bank of senior one
Given that f (x) is a quadratic function, the vertex of its image is (1,3) and passes through the origin, we can find f (x)

Y = ax ^ 2 + BX + C (a ≠ 0), passing through the origin, x = 0, y = 0 and substituting, we can get C = 0, the vertex, that is, the maximum value (1,3), - B / 2A = 1, (4ac-b ^ 2) / 4A = 3, because C = 0, we can get a = - 3, B = 6, (b = 0 rounding), that is, f (x) = - 3x ^ 2 + 6x

The function defined on R is y = f (x), f (0) ≠ 0, when x ＞ 0, f (x) ＞ 1, and for any a, B ∈ R, f (a + b) = f (a) f (b), (1) proof: F (0) = 1; (2) proof: for any x ∈ R, f (x) ＞ 0; (3) given that f (x) is an increasing function on R, if f (x) · f (2x-x2) ＞ 1, find the value range of X

(1) Let a = b = 0, then f (0) = [f (0)] 2 ∵ f (0) ≠ 0 ∵ f (0) = 1 (2) let a = x, B = - x, then & nbsp; f (0) = f (x) f (- x) ∵ f (- x) = 1F (x) from known x > 0, f (x) > 1 > 0, when x < 0, - x > 0, f (- x) > 0 ∵ f (x) = 1F (- x) > 0 and x = 0, f (0

Given that f (x) satisfies f (f (x)) = 4x + 3, the analytic expression of F (x) is obtained

Let f (x) = ax + B, then f (f (x)) = a (AX + b) + B = a2x + AB + B = 4x + 3,  A2 = 4AB + B = 3, a = 2B = 1 or a = − 2B = − 3, ｜ f (x) = 2x + 1, or F (x) = - 2x-3

Let a be a real number, f (x) = x2 + | x-a | + 1, X ∈ R (1) if f (1) = 2, find the value of a; (2) discuss the parity of F (x); (3) find the minimum value of F (x)

(1) ∵ f (1) = 2, f (x) = x2 + | x-a | + 1, ∵ 1 + | 1-A | + 1 = 2, a = 1. (2) for the function F (x) = x2 + | x-a | + 1, when a = 0, f (x) = x2 + | x | + 1 is even function, when a ≠ 0, f (x) = x2 + | x | + 1 is non odd and non even function. (3) when x ≤ a, f (x) = x2-x + A-1 = (x − 12) 2 + A + 34, if a > 12, the minimum value of function f (x) is f (12) = a + 34; if a ≤ 12, the minimum value of function f (x) is f (a) = A2 + 1 If a ＞ - 12, the minimum value of function f (x) is f (a) = A2 + 1; if a ≤ - 12, the minimum value of function f (x) is f (- 12) = - A + 34

If f (x) > 0 and f (x · y) = f (x) · f (y), if x > 1, then f (x) > 1
(1) Find f (1) (to be 1)
(2) It is proved that the function f (x) is a monotone increasing function on (0, + ∞)
(3) It is proved that the function f (x) is even
(4) Solving inequality f (X-2) - f (2x-1) < 0

1.
Set 01
f(x2)>f(x1)
two
f(x^2)=f(x)^2=f(-x)^2
f(x)>0
f(x)=f(-x)
three
f(x-2)

There are 20 workers in an automobile parts manufacturing workshop. It is known that each worker can make 6 class a parts or 5 class B parts every day, and the profit is 150 yuan for each class a part and 260 yuan for each class B part. The workshop arranges x workers to make class a parts every day, and the rest workers to make class B parts. (1) please write down the profit y (yuan) of this workshop every day ）(2) how many workers do you think it is appropriate to send at least to manufacture type B parts if the profit of the workshop is not less than 24000 yuan per day?

(1) According to the meaning of the title, we can get y = 150 × 6x + 260 × 5 (20-x) = - 400X + 26000 (0 ≤ x ≤ 20); (2) from the meaning of the title, we know that y ≥ 24000, that is - 400X + 26000 ≥ 24000, let - 400X + 26000 = 24000, we can get x = 5, At this time, there are 20-x = 20-5 = 15. A: at least 15 workers should be sent to make type B parts

Two straight lines y = 3x y = - 2x + 3
The area enclosed in a triangle on the y-axis is

The abscissa of the intersection of two lines is 3x = - 2x + 3
The solution is x = 0.6
The height of the triangle is 0.6
It is known that the base length of triangle is 3
So the area of the triangle is 0.9

Several mathematical function problems
1. Given that the parabola y = x ＾ 2 + X + B ＾ 2 passes through points (a, negative quarter) and (- A, Y1), what is Y1?
2. It is known that the image of quadratic function y = ax ＾ 2 + BX + C intersects X-axis at two points a and B, intersects Y-axis at point C, and triangle ABC is a right triangle. Please write a quadratic function that meets the requirements
3. If the parabola y = x ＾ 2 + BX + C intersects with the positive half axis of X axis at two points a and B, intersects with y axis at point C, and the length of line AB is 1, and the area of triangle ABC is 1, then the value of B is

1. According to the meaning of the title, the first point is brought into the parabola analytical expression
-1/4=a^2+a+b^2 ①
So Y1 = a ^ 2-A + B ^ 2
① (2) Y1 = - 1 / 4-2a
2. If you want ABC to be a right triangle, you can see that angle c is a right angle. If the origin is O, then angle OCA and angle OCB and angle OAC and angle OBC are 45 degrees, then OA = ob = OC, when x = 0, OC = C, when y = 0, OA = ob
The quadratic function y = - x ^ 2 + 1 can be obtained
3. The best way to solve this problem is to draw a picture. The area of triangle ABC is equal to ab * OC, because AB = 1, the area of triangle ABC is equal to 1, so 1 / 2 * 1 * OC = 1, so OC = 2, that is, when x = 0, y = C = + 2 or - 2, but the parabola has two intersections with the positive half axis of X, so C = 2, and then B = + 3 or - 3, or because there are two intersections with the positive half axis of X, so B = - 3

1. In the following problems, the one in which two variables do not form a functional relationship is ()
A. 2x-1 and X
B. The square root of a (a ≥ 0) and a
C. Radius and area of circle
D. How much electricity do you pay each month
2. The perimeter of a rectangle is 12cm. Find the relationship between its area s (cm ^ 2) and its side length x (CM), and find out the value range of X

1 B because the square root with positive and negative is equal to 2, the solution is not a function. Everything else can be calculated
2 s = 6x-x square (0