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It is known that the quadratic function f (x) = ax Λ 2 + BX + 1 (a, B are constants, a ＞ 0) satisfies f (x-3) = f (1-x) and has the maximum value 4 in the interval ＜ - 3,2 ＞ find the value of a, B and find the range of the function f (x) in the interval ＜ - 3,2 ＞
F (x-3) = f (1-x), so the axis of symmetry is [x-3 + 1-x] / 2 = 2, so - B / 2A = 2, and because there is a maximum value of 4 on the interval ＜ - 3,2 ＞, and in this interval, f (- 3) = 4, so we get 9a-3b + 1 = 4, so we get a = 1 / 7, B = - 4 / 7, because the interval ＜ - 3,2 ＞ is simple
The opening of integral coefficient quadratic function y = ax ^ 2 + BX + C is upward, and the discriminant is B ^ 2-4ac = 5. The values of a, B and C are calculated between the two intersections of X axis and the interval (- 1,0), (1, + ∞)
x1=(-b-√5)/2a.-10
-2a
Given the following equation: √ 9 * 9 + 19 = 10, √ 99 * 99 + 199 = 100, √ 999 * 999 + 1999 = 1000, write the nth equation
√ 99 (n) 9 * 99 (n) 9 + 199 (n) 9 = 10 & # 8319;
According to the above characteristics, write the fourth formula and verify it
√(9999*9999+19999)=10000
Let a = 9999
Then 19999
=9999+9999+1
=2a+1
So the original formula = √ (A & # 178; + 2A + 1)
=√(a+1)²
=a+1
=10000
RELATED INFORMATIONS
- 1. The quadratic function f (x) = ax ^ 2 + BX + C satisfies the following conditions 1) F (- 1) = f (- 3); 2) f (x) has a minimum value of 3; 3) f (x) image passes through point (0,4), find the analytic expression of function f (x)
- 2. It is known that the left focus of the ellipse x ^ 2 + y ^ 2 / b ^ 2 = 1 (b belongs to (0,1)) is f, the left and right vertices are AC, the upper vertex is B, and the circle P is made through three points of FBC, where the coordinates of the center P are (m, n). Let f, B and C be (- C, 0), (0, b), (1,0) respectively. If the eccentricity of the ellipse e = 2 / 3, the equation of the circle P is obtained
- 3. It is known that the right focus a of ellipse C: x ^ 2 / A ^ 2 + y ^ 2 / b ^ 2 = 1 (a > b > 0) is the focus of parabola y ^ 2 = 8x, the vertex on the circle is B, and the eccentricity is √ 3 / 2,1, Find the path of ellipse C; 2, the line L passing through point (0, √ 2) with slope k intersects ellipse C at two points P and Q. if the abscissa of the midpoint of line PQ is 4 √ 2 / 5, find the equation of line L
- 4. The center of a circle is on the right focus of the ellipse 16x ^ 2 + 25y ^ 2 = 400, and passes through the vertex of the ellipse on the y-axis to find the equation of the circle
- 5. The two ends of the minor axis of the ellipse form 120 degrees with the line connecting the focus, and the eccentricity of the ellipse is calculated
- 6. From an end point of the ellipse minor axis to see the two focal points, the angle of view is 120 to calculate the eccentricity
- 7. (2011 · Anhui simulation) the area of △ pf1f2 is () A. 20B. 22C. 24D. 28
- 8. The chord passing through the focus of the parabola and perpendicular to the axis of symmetry is called the path of the parabola. Find the equation of the parabola whose vertex is at the origin and the path length is 8 Because the topic didn't give the focus position, it was discussed in four situations . How do we get 2p = 8?
- 9. The focus coordinates of the ellipse X25 + y24 = 1 are______ ．
- 10. If a line passing through a focus F1 of ellipse 4x2 + 2Y2 = 1 intersects with ellipse at two points a and B, then a and B and another focus F2 of ellipse form △ abf2, then the perimeter of △ abf2 is () A. 2B. 22C. 2D. 1
- 11. Y = (m-2) x to the power of M + 1 + 3x is a positive proportion function, and finding the relationship of positive proportion function is a formula
- 12. If the point P (a, √ 3) is on the positive scale function y = - 3x, find the value of A
- 13. It is known that in the quadratic function y = AX2 + BX + C, the corresponding values of function y and independent variable X are shown in the following table X - 3 - 20135 y 708 - 9 - 57 find the function value y when the symmetry axis X = 2 of y = ax + BX + C
- 14. It is known that the quadratic function y = AX2 + BX + C (a ≠ 0) satisfies the following quantitative relationship between the independent variable x and the function value y X negative two thirds negative one negative half 0 half 1 half three Four fifths of Y minus two minus nine fourths of Y minus two minus five fourths of Y minus seven fourths of Y
- 15. The image of the linear function Y1 = KX + B and the inverse scale function y2 = m / X intersect at two points a (- 2,1), B (1, n) 1) Find the expression of first-order function and inverse proportion function 2) Write Y1 > Y2
- 16. The function f (x) = X3 + AX2 + BX + C, where a, B and C are real numbers. When a2-3b < 0, f (x) is () A. Increasing function B. decreasing function C. constant D. is neither increasing function nor decreasing function
- 17. It is known that the minimum value of function f (x) = (2 ^ x-a) ^ 2 + (2 ^ - x-a) ^ 2 is 8, and there is a value range of real number a
- 18. It is known that a and B are real numbers, f (x) = x ^ 3 + ax, G (x) = x ^ 2 + BX, f '(x) and G' (x) are derivatives of F (x), G (x), if f '(x) g' (x) ≥ 0 in function area If f (x) and G (x) are monotonically consistent on interval I, then f (x) and G (x) are monotonically consistent on interval I 1. Let a > 0, if the monotonicity of functions f (x) and G (x) is consistent in the interval [- 1, + ∞), find the value range of real number B
- 19. Let the asymptote of hyperbola x2a2 − y2b2 = 1 (a > 0, b > 0) be tangent to the parabola y = x2 + 1, then the eccentricity of the hyperbola is equal to () A. 3B. 2C. 5D. 6
- 20. It is known that a focus of hyperbola C: x ^ 2 / A ^ 2-y ^ 2 / b ^ 2 = 1 (a > 0, b > 0) is F2 (2,0) and B = root 3a. (1) the equation for solving hyperbola C (2) let a normal vector of line L passing through focus F2 be (m, 1). When line L and the right branch of hyperbola C intersect at two different points a and B, then, Find the value range of the real number m: and prove that the key point m of AB is on the curve 3 (x-1) ^ 2-y ^ 2 = 3. (3) let the right branch of the straight line L and the hyperbola C in (2) intersect at two points a and B. ask if there is a real number m, so that the angle AOB is an acute angle? If there is, find the range of M. if not, explain the reason