若不等式2x-1>m(x^2-1)对满足│m│

1个回答

  • 构造变量m的函数求

    2x-1>m(x2-1)即:(x^2 - 1)*m - (2x - 1) < 0

    构造关于m的函数f(m) = (x^2 - 1)*m - (2x - 1)

    1)当x^2 - 1 > 0时, 则f(2) < 0

    从而 2x^2 - 2x - 1 < 0

    解得: (1 - √3) / 2 < x < (1 + √3) / 2

    又x^2 - 1 > 0,即x < -1 或 x > 1, 所以 1 < x < (1 + √3) / 2;

    2)当x^2 - 1 < 0时, 则f(-2) < 0 => -2x^2 - 2x + 3 < 0

    从而 2x^2 + 2x - 3 > 0

    解得 x < (-1 - √7) /2 或x > (√7 - 1) / 2

    又-1 < x < 1, 从而(√7 - 1) / 2 < x < 1

    3)当x^2 - 1 = 0时, 则f(m) = 1 - 2x < 0 从而x > 1/2,故x = 1;

    综上有:(√7 - 1)/2 < x < (1 + √3) / 2