If f (x) = AX2 + BX, and 1 ≤ f (1) ≤ 2,2 ≤ f (1) ≤ 4, find the range of F (2)
From F (x) = ax ^ 2 + BX, we can get f (- 1) = A-B, ① f (1) = a + B, ② f (- 2) = 4a-2b, from ① + ② we can get a = [f (1) + F (- 1)], from ② - ① we can get b = [f (1) - f (- 1)], thus f (- 2) = 2 [f (1) + F (- 1)] - [f (1) - f (- 1)] = 3f (- 1) + F (1) and 1 ≤ f (1) ≤ 2,2
RELATED INFORMATIONS
- 1. If f (x) = AX2 + BX (a ≠ 0) satisfies 1 ≤ f (- 1) ≤ 2, 2 ≤ f (1) ≤ 5, then the value range of F (- 3) is______ .
- 2. 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
- 3. 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
- 4. 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)
- 5. 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
- 6. 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
- 7. Given the quadratic function f (x) = AX2 + BX + C, f (2) = 0, f (- 5) = 0, f (0) = 1, find the quadratic function
- 8. If the quadratic function f (x) = AX2 + BX + C, if f (0) = 0 and f (x + 1) = f (x) + X + 1, then f (x)=
- 9. 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
- 10. 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
- 11. Let f (x) = alnx-1 / 2x ^ 2 + BX find the solution set of the inequality f (x) > F (1)
- 12. 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)
- 13. The function f (x) = alnx-bx ^ 2 (x ≥ 0) when B = 0, if the inequality f (x) ≥ m + X holds for all a ∈ [0,3 / 2], X ∈ (1, e ^ 2), find the range of M
- 14. What is the domain of (1-e ^ x) under the function f (x) = 1 / radical
- 15. What is the domain of 1-e ^ x under the function f (x) = 1 / radical
- 16. Function y = - x2-2ax (0
- 17. If the maximum value of function f (x) = x & sup2; + 2aX + 1 on [0,1] is f (1), then what is the value range of a
- 18. Finding the minimum value of the square of function y = x-2ax-1 on [0,2]
- 19. F (x) = x ^ 2-2ax + 2. X ∈ [- 1.1] to find the minimum value of function
- 20. It is known that y = 2x * x-2ax + 3 has a minimum value on [- 1,1], which is denoted as G (a). Find the expression of the function and its maximum value