If the quadratic function f (x) = AX2 + BX + C, if f (0) = 0 and f (x + 1) = f (x) + X + 1, then f (x)=
f(0)=0+0+c=0
So C = 0
f(x+1)=a(x+1)²+b(x+1)
=ax²+(2a+b)x+(a+b)
=f(x)+x+1
=ax²+(b+1)x+1
therefore
2a+b=b+1
a+b=1
So a = b = 1 / 2
f(x)=x²/2+x/2
RELATED INFORMATIONS
- 1. It is known that the quadratic function f (x) = ax ^ 2 + BX satisfies f (2) = 0 and the equation f (x) = x has equal roots
- 2. It is known that the quadratic function f (x) = ax ^ 2 + BX (AB ∈ R, a ≠ 0) satisfies f (- x + 5) = f (x-3) and the equation f (x) = x has equal roots; It is known that the quadratic function f (x) = ax ^ 2 + BX (AB ∈ R, a ≠ 0) satisfies f (- x + 5) = f (x-3) and the equation f (x) = x has equal roots. (1) find the analytic expression of F (x); (2) whether there are real numbers m, n (m < n), so that the domain of definition and value of F (x) are [M, n] and [3M, 3N]? If there are, find the value of M, N; if not, explain the reason
- 3. It is known that the quadratic function f (x) = ax + BX (a, B are familiar, and a ≠ 0) satisfies the following conditions: F (- x + 5) = f (x-3), and the equation f (x) = x has equal roots (1) The expression of finding f (x) (2) whether there is a teacher's uncle m, n (m) or not
- 4. It is known that f (x) is an odd function defined on R. when x ≥ 0, f (x) = x2-2x, then the expression of F (x) on R is () A. y=x(x-2)B. y=x(|x|-1)C. y=|x|(x-2)D. y=x(|x|-2)
- 5. The problem of derivative in second grade mathematics Find the tangent equation of the curve y = SiNx at point a (6 / 6 π, 2 parts only 1)? 2. Find the tangent equation of the square of the parabola y = x parallel to the line 2x-y = 0? 3. Find the tangent equation of the point (0,0) and the square of the curve y = x? 4. Find the tangent equation of the point (1, - 1) on the curve y = XD 3 power - 2x? To write the process, thank you
- 6. Is derivative the instantaneous rate of change Just learning derivative, it's a bit abstract for me, I can't help but be grateful!
- 7. Take a point (1,2) and a nearby point (1 + △ x, 2 + △ y) on the image of curve y = x2 + 1, then △ y △ x is______ .
- 8. How to understand the instantaneous change rate of point derivative? When the increment of the independent variable approaches the limit of 0, that is, the instantaneous rate of change of the point derivative of the point, the problem is why the increment of the independent variable is still meaningful when it is all 0, and why there is still a rate of change when there is no change at the point?
- 9. F (x) = [(x ^ 2 under the third radical) - (x under the second radical)] divide by X (x > = 0) to find the limit of F (x) at x = 0 F (x) = [(x ^ 2 under the third radical) - (x under the second radical)] divide by X (x > = 0) to find the limit of F (x) at x = 0
- 10. The limit of F (x) is a, a > 0. It is proved that f (x) is equal to a under the triple root sign Can be added!
- 11. Given the quadratic function f (x) = AX2 + BX + C, f (2) = 0, f (- 5) = 0, f (0) = 1, find the quadratic function
- 12. Quadratic function, f (x) = AX2 + BX + C, f F (x) = AX2 + BX + C, f (x) = f (2-x), f (0) = 3, f (1) = 2.1. Find the analytic formula of function. 2. F (x) belongs to the maximum value of [- 1,2] in X
- 13. It is known that the quadratic function f (x) = AX2 + BX + C (a > 0) has two different common points with the X axis If f (c) = 0 and 0 < x < C, f (x) > 0 1) Compare 1 / a with C 2) Proof - 2 < B < 1
- 14. Given the quadratic function f (x) = AX2 + BX + C (C ≠ 0) (1) if A.B.C and f (1) = 0, it is proved that the image of F (x) has two intersections with the X axis; (2) if Changshu x1 If x 2 ∈ R, and x 1, x 2, f (x 1) ≠ f (x 2), we prove that the equation f (x) = 1 / 2 [f (x 1) + F (x 2)] must have a root belonging to (x 1, x 2)
- 15. 7. Given the quadratic function f (x) = AX2 + BX + C, G (x) = λ ax + B (λ≥ 1), when | x | ≤ 1, | f (x) | ≤ 1. (1) it is proved that | a | ≤ 2 (2) Let f (0), f (1), f (- 1) denote g (1), G (- 1) (3) When | x | ≤ 1, it is proved that | g (x) | ≤ 2 λ
- 16. It is known that the derivative f '(x) ≤ f (x) B of quadratic function f (x) = AX2 + BX + C belongs to R. it is proved that f (x) is less than or equal to (x + C) ^ 2 when x ≥ 0
- 17. If f (x) = AX2 + BX (a ≠ 0) satisfies 1 ≤ f (- 1) ≤ 2, 2 ≤ f (1) ≤ 5, then the value range of F (- 3) is______ .
- 18. If f (x) = AX2 + BX, and 1 ≤ f (1) ≤ 2,2 ≤ f (1) ≤ 4, find the range of F (2)
- 19. Let f (x) = alnx-1 / 2x ^ 2 + BX find the solution set of the inequality f (x) > F (1)
- 20. Let f (x) = alnx − 12x2 + BX. (1) when a = 3, B = 12, find the maximum value of F (x); (2) find the solution set of the inequality f '(x) > F (1)