Given the function y = y1-y2, where Y1 and X are in positive proportion, Y2 and X are in inverse proportion, and when x = 1, y = 1; when x = 3, y = 5

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• 1. 1. Given the function Y1 = x (x > 0) and function y2 = 1 / X (x > 0), then=____ The minimum value of Y1 + Y2 is____ . 2. Given function Y1 = x + 1 (x > - 1) and function y2 = (x + 1) & # + 4 (x > - 1), then=____ The minimum value of Y2 / Y1 is___ .
• 2. If the quadratic functions F1 (x) = a1x2 + b1x + C1 and F2 (x) = a2x2 + b2x + C2, let F1 (x) + F2 (x) be an increasing function on (- ∞, + ∞) if______ ．
• 3. If the quadratic functions F1 (x) = a1x ^ 2 + b1x + C1 and F2 (x) = a2x ^ 2 + b2x + C2 such that F2 (x) + F1 (x) is listed as an increasing function in (- ∞, + ∞), the condition is
• 4. If the quadratic function F1 (x) - a1x ^ 2 + b1x + C1, F2 (x) = a2x ^ 2 + b2x + C2, such that F1 (x) + F2 (x) is an increasing function on R, please give the following result If the quadratic function F1 (x) - a1x ^ 2 + b1x + C1, F2 (x) = a2x ^ 2 + b2x + C2, such that F1 (x) + F2 (x) is an increasing function on R, please give a set of F1 (x) =? F2 =?
• 5. If the quadratic functions f 1 (x) and F 2 (x) satisfy the following conditions: (1) f (x) = f 1 (x) + F 2 (x) monotonically decreases on R; (2) g (x) = f 1 (x) - F 2 (x) pairs (2) For any real number x1, X2 (x1 ≠ x2), if G (x1) + G (x2) / 2 > G (x1 + x2 / 2), then F1 (x) = F2 (x)=
• 6. As shown in the figure, in the plane rectangular coordinate system xoy, the analytical expression of the parabola is
• 7. The parabola y = - 1 / 2x square + 3 / 2x + 2 intersects the X axis at two points a and B, and intersects the Y axis at point C (1) ABC is a right triangle (2) : the straight line x = m (0 ∠ m ∠ 4) moves on the line ob, intersects the x-axis at point D, intersects the parabola at point E, intersects BC at point F. when m = what, EF = DF? (3) : after connecting CE and be, "is there a point e to maximize the area of triangle BCE?" if there is a point E, calculate the coordinates of point E and the maximum area of triangle BCE
• 8. In the plane rectangular coordinate system, the line L and the parabola y ^ 2 = 2x intersect at two points a and B (1) Prove: "if the line L passes through the point t (3,0), then the product of vector OA and vector ob = 3" is a true proposition (2) Write down the inverse proposition of the proposition in (1), judge the truth and explain the reason
• 9. In the plane rectangular coordinate system, the intersection of the parabola Y1 = x ^ 2-2x + A and the X axis is a, and the intersection of the parabola y2 = x ^ 2 + 2x + 1 + 2a and the X axis is B, and B is symmetric about the Y axis, a is a real number (1) Calculate the value of a and the coordinates of a and B; (2) Do the parabola Y1 and Y2 intersect at a point C on the y-axis? If they intersect at the same point, ask for the area of the largest triangle ABC. If they do not intersect at the same point, please explain the reason
• 10. Given that the points a (a, Y1), B (2a, Y2), C (3a, Y3) are all on the parabola y = 1 / 2x ^ 2-1 / 2x, find the intersection of the parabola and X-axis? How to find the area of triangle ABC when a = 1? 3. Is there any equation which contains Y1, Y2, Y3 and has nothing to do with a? If so, try to give one and prove it: if not, explain the reason?
• 11. If we know that the two tangents of parabola C1: y2 = x + 1 and parabola C2: y2 = - x-a are perpendicular at the intersection, we can find a
• 12. It is known that line L and parabola C1: y = - X2, C2: y = - x2 + ax are tangent to points a and B respectively, and Given that the line L is tangent to the parabola C1: y = - X2, C2: y = - x2 + ax respectively, and points a, B, "ab" = 3 root sign, 5 divided by 4, find the value of A
• 13. Given that parabola C1: y = 2x2 and parabola C2 are symmetric with respect to line y = - x, then the Quasilinear equation of C2 is () A. x=18B. x=-18C. x=12D. x=-12
• 14. Given the image of parabola C1: y = x ^ 2-2x as shown in the figure, fold the image of C1 along the Y axis to get the image of parabola C2 Given the image of parabola C1; y = x ^ 2-2x as shown in the figure, fold the image of C1 along the Y axis to get the image of parabola C2. 1) if the line y = x + B and parabola y = ax ^ 2 + BX + C (a is not equal to 0) have and only have one intersection, the line is said to be tangent to the parabola. If the line y = x + B is tangent to parabola C1, the value of B is calculated 2) When the line y = x + B and the image C1, C2 have two intersections, the value range of B
• 15. The curve C1: y = x ^ 2yu and C2: y = - (X-2) ^ 2, the line L and C1C2 are tangent, find the equation of the line L. use the derivative function of two curves to find out why it can't be used
• 16. Given the curve y = - x ^ 3 + 2x, find the linear equation which passes through point B (2,0) and is tangent to curve C? Derivative = -= I set a point (x0, Y0), and then f (x0 + △ x) - f (x0) / △ x, want to find its derivative function = = but it's so strange, is it - △ x ^ 2-3x0 ^ 2-3 △ xX0 + 1, is it my method problem or wrong calculation? = = because I feel that △ x should all be reduced
• 17. As shown in Figure 11, parabola and straight line y = kx-4k (k)
• 18. If the parabola y = (K + 1) x2 + k2-9 has a downward opening and passes through the origin, then K=______ ．
• 19. It is known that there are two different points e (K + 3, - K2 + 1) and f (- k-1, - K2 + 1) on the parabola y = - x2 + BX + 4 It is known that there are two different points e (K + 3, - K2 + 1) and f (- k-1, - K2 + 1) on the parabola y = - x2 + BX + 4 (1) Find the analytical formula of parabola; (2) As shown in the figure, the parabola y = - X & # 178; + BX + 4 intersects with the positive half axis of X axis and Y axis at points a and B respectively, M is the midpoint of AB, PMQ rotates with m as the center on the same side of AB, and PMQ = 45 °, MP intersects with y axis at point C, MQ intersects with X axis at point D; (3) Under the condition of (2), when the values of M and N are, the edge of ∠ PMQ passes through the point F?
• 20. Square of parabola y = x, section line y = x, the length of line segment is