Given the function f (x) = x + 4 / x, judge the monotonicity of F (x) on (0,2) and (2, positive infinity) and prove it

Given the function f (x) = x + 4 / x, judge the monotonicity of F (x) on (0,2) and (2, positive infinity) and prove it

f(x)=x+4/x
f'(x)=1-4/x^2
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F (x) increase

Given the function f (x) = x / (x + 1), if the sequence {an} (n ∈ n *) satisfies: A1 = 1, an + 1 = f (an), find the general formula of {one of an}

f(x)=x/(x+1)
a(n+1)=an/(an+1)
1/a(n+1)=(an+1)/an=1+1/an
d=1/a(n+1)-1/an=1
1/an=1/a1+(n-1)d=n

If the image of function y = ax ^ 2 + BX + C passes through the origin, what is the necessary and sufficient condition?

Because the image passes through the origin, the origin coordinates (0,0) satisfy this functional equation
Namely: 0 = a * 0 ^ 2 + b * 0 + C
0=0+0+c
So C = 0

If and only if the function y = AX2 + BX + C (a ≠ 0) passes through the origin______ ．

If the function y = AX2 + BX + C (a ≠ 0) passes through the origin, that is, O (0, 0) is on the image, and (0, 0) is replaced by the analytic expression, then C = 0. Conversely, if C = 0, then y = AX2 + BX, when x = 0, y = 0, that is, y = AX2 + BX + C (a ≠ 0) passes through the origin, so the answer is: C = 0

Note that the sum of the first n terms of the sequence {an} is Sn, and Sn = 2 (an-1), then A2=______ ．

∵ Sn = 2 (an-1), ∵ S1 = 2 (A1-1), ∵ A1 = 2 ∵ S2 = 2 (A2-1) = 2 + A2 ∵ A2 = 4, so the answer is: 4

The sequence {an} is an arithmetic sequence, A1 = f (x + 1), A2 = 0, A3 = f (x-1), where f (x) = x square -- 4x + 2?

The arithmetic sequence: a2-a1 = a3-a2, A2 = 0, so: a1 + a3 = 0, expand f (x + 1) and f (x-1) into a1 + a3 = 0, there are: x ^ 2-4x + 3 = 0, so when x = 1 or x = 3, x = 1, substitute A1 = f (x + 1) to get A1 = - 2, a2-a1 = 2 = D. when an = - 2 + 2 (n-1) x = 3, substitute A1 = f (x + 1) to get A1 = 2, a2-a1 = - 2 = D

The sequence {an} is an arithmetic sequence, A1 = f (x + 1), A2 = 0, A3 = f (x-1), where f (x) = x2-4x + 2, then the general term formula an = ()
A. - 2n + 4B. - 2n-4c. 2n-4 or - 2n + 4D. 2n-4

∵ f (x) = x2-4x + 2, ∵ A1 = f (x + 1) = (x + 1) 2 − 4 (x + 1) + 2 = x2-2x-1, A3 = f (x − 1) = (x − 1) 2 − 4 (x − 1) + 2 = x2-6x + 7, and the sequence {an} is an arithmetic sequence, A2 = 0 ∵ a1 + a3 = 2A2 = 0, ∵ (x2-2x + 1) + (x2-6x + 7) = 2x2-4x6 = 0, the solution is: x = 1 or x = 3 & nbsp; & nbsp

Given that f (x) = x ^ 2-4x + 3, in the arithmetic sequence {an}, A1 = f (x-1), A2 = - 1 / 2, A3 = f (x), find the general term formula of the sequence {an}

If we know a1 + A2 = - 1 and f (x) = x ^ 2-4x + 3, we can get x = 2 or 3. When x = 2, A1 = 0a2 = - 1 / 2. A3 = - 1, then an = 1 / 2-1 / 2n
When x = 3, A1 = - 1, A2 = - 1 / 2, A3 = 0, then an = 1 / 2n-3 / 2

If A1 = f (x + 1), A2 = 2, A3 = f (x-1), where f (x) = 3x-2, find the general term formula an

According to the meaning of the title, A1 = f (x + 1) = 3 (x + 1) - 2 = 3x + 1,
a3=f(x-1)=3(x-1)-2=3x-5,
From {an} is an arithmetic sequence, we can see that a1 + a3 = 2A2, that is, 3x + 1 + 3x-5 = 2 × 2, and the solution is x = 4 / 3
So A1 = 5, tolerance = a2-a1 = - 3, an = 5 + (n-1) × - 3 = - 3N + 8

The function f (x) = a1x + a2x ^ 2 +. + anx ^ n, A1, A2, A3,... An is an arithmetic sequence, n is a positive even number, and f (1) = n ^ 2, f (- 1) = n, find an

F (1) = a1 + A2 +... + an = n ^ 2 Sn = n (a1 + an) / 2 = n ^ 2 = > (a1 + an) / 2 = [A1 + A1 + (n-1) D] / 2 = n... 1 formula F (- 1) = - a1 + A2-A3 +... + an = (a2-a1) + (a4-a3) +... + [an-a (n-1)] n is a positive even number = > nd / 2 = n = > d = 2 into 1 formula = > A1 = 1 substitute A1 d into 1 formula = > 1 + an = 1 + 1 +