It is known that if the function FX defined on R satisfies f (x + 2) = - 1 / F (x), and X belongs to [0,1], FX = 2x + 1 Then, f8.5 is?

It is known that if the function FX defined on R satisfies f (x + 2) = - 1 / F (x), and X belongs to [0,1], FX = 2x + 1 Then, f8.5 is?

Given that the function FX defined on R satisfies f (x + 2) = - 1 / F (x), and FX = 2x + 1 when x belongs to [0,1], then f8.5 is?
Analysis: ∵ the function FX defined on R satisfies f (x + 2) = - 1 / F (x),
Let x = x + 2 be substituted into f (x + 4) = - 1 / F (x + 2) = f (x)
The function f (x) is a periodic function with the minimum positive period of 4
When x belongs to [0,1], FX = 2x + 1
f(8.5)=f(0.5+2*4)=f(0.5)=2

If the function f (x) = ax ^ 4 + BX ^ 2 + C is known and the tangent equation at x = 1 is y = X-2, then the analytic expression of F (x) is given

If we take the point (0,1) into the equation, we have
1=c
Take x = 1 into tangent sum function equation and get tangent point
(1,a+b+c)=(1,-1)
The slope is obtained by deriving the function from X
k=4ax^3+2bx=4*a*1+2*b*1=4a+2b =1
It can be obtained by combining the above equations
a=2.5 b=-4.5 c=1
So the original function is f (x) = 2.5x ^ 4-4.5x ^ 2 + 1

Given the function f (x) = ax ^ 3 + BX ^ 2 + C, the image passes through (0,1) and the tangent equation at x = - 2 is 2x + y + 2=
=0 to find the monotone decreasing interval of the analytic expression y = f (x)

The image of known function f (x) = ax ^ 3 + BX ^ 2 + C is obtained by (0,1)
c=1
Given the function f (x) = ax ^ 3 + BX ^ 2 + C, the tangent equation at x = - 2 is 2x + y + 2 = 0
F '(x) = 3ax ^ 2 + 2bx = - 2, that is 12a-8b = - 2
If x = - 2, y = 2, then 2 = - 8A + 4B + 1
A = 0, B = 1 / 4
The analytic formula of y = f (x) is y = 0.25x ^ 2 + 1
Monotone decreasing interval is (- ∞, 0)

It is proved that f (x) = cos ^ 2x + cos ^ 2 (x + Π / 3) + cos ^ 2 (x - Π / 3) is a constant function

cos^2(x+∏/3)+cos^2(x-∏/3)
=(cosx / 2-radical 3 * SiNx / 2) ^ 2 + (cosx / 2 + radical 3 * SiNx / 2) ^ 2
=(cosx)^2/2+3(sinx)^2/2
=1/2+(sinx)^2
f(x)=cos^2x+cos^2(x+∏/3)+cos^2(x-∏/3)
=(cosx)^2+1/2+(sinx)^2
=3/2,
It's a constant. It's proved

Let f (x) = ln (a + x ^ 2) x > 1 = x + B X

As shown in the picture & nbsp; I'm sorry to send the wrong one

The function f (x) = ln (x + 1) - ax + (1-A) / (x + 1) (a > 0.5) (1) when the tangent of the curve y = f (x) at (1, f (x)) is perpendicular to the straight line L: y = 2X + 1
Given the function f (x) = ln (x + 1) - ax + (1-A) / (x + 1) (a > 0.5) (1) when the tangent of the curve y = f (x) at (1, f (x)) is perpendicular to the straight line L: y = 2x + 1, find the value of a (2) find the monotone interval of function f (x) (3) prove: (1 / 2) + (1 / 3) + (1 / 4) + +[1/(n+1)]

1. F (x) derivative of 1 / (1 + x) - A - (1-A) / ((x + 1) * (x + 1)), substituting x = 1, the slope is 0.25-0.75 * a, and the product of 2 is - 1, so a = 1;
The derivative is reduced to a / (t-1, t-1), and the rest is reduced to a / (t-1, t-1)

Given the circle: X * 2 + y * 2-6x-8y = 0, make a chord of length 6 through the origin of the coordinate, then the equation of the line where the chord is located is

Transform the equation of circle: (x-3) ^ 2 + (y-4) ^ 2 = 25
So if the center of the circle is (3,4), the radius is 5, and the chord length is 6, then the distance from the center of the circle to the straight line is 4
Let the linear equation be y = KX, and then calculate with the distance formula
The equation is y = 0 or 24x + 7Y = 0

Given the circle C: X & # 178; + Y & # 178; - 2x-4y-11 = 0, then in the straight line passing through the origin, the sum of the lengths of the longest chord and the shortest chord cut by circle C is
A. 8 + 2 times root 11
B. 8 + 4 times root 11
C. 4 + 2 times root 11
D. 4 + times root 11

In the line passing through the origin, the longest chord cut by circle C is the diameter passing through the origin, and the shortest chord is the chord perpendicular to the diameter and passing through the origin
The sum of its length is a, that is 8 + 2 times the root 11

Given that C: x ^ 2 + y ^ 2-10x = 0, the chord length of the straight line L passing through the origin cut by the circle C is 8, find the hyperbolic equation with the circle C as the center and l as the asymptote

Let l be y = KX, and circle C be reduced to the standard formula: (X-5) + y = 25. Then according to the relationship between chord center distance, chord length and radius, we can find out that the distance between chord center and circle center is the distance from circle center to chord, which can be calculated according to the distance from point to line: D = 5K / √ (K + 1) according to: D + 4 = 25, we can find out k = ± 3 / 4 〈 L: y = ± 3 / 4 in hyperbola with C as focus and l as asymptote: C = 5, B / a = 3 / 4 according to hyperbola: a + B = C, So the hyperbolic equation is: X / 16-y / 9 = 1

Given the circle C: x2 + y2-10x = 0, the chord length of the line L passing through the origin cut by the circle C is 8, and the hyperbolic equation of the asymptote is obtained

Let l be y = KX, and circle C be reduced to the standard formula: (X-5) ² + Y & #178; = 25. Then, according to the relationship between chord center distance, chord length and radius, we can find out that K chord center distance is the distance from the center of the circle to the chord, which can be calculated according to the distance from the point to the line: D = 5K / √ (K & #178; + 1) according to: D & #178; + 4 & #178; = 25, we can find out k = ± 3 / 4  L: