Решите неравенство x^(_2 sqrt(x)) (2)/(sqrt(x)) .

Задача №17574: Неравенство

Решите неравенство $x^{l o g_{2} x} ⩾ \frac{2}{x} .$

x^{l o g_{2} x} ⩾ \frac{2}{x} ⟺ 2^{1/2 l o g_{2}^{2} x} ⩾ 2^{1 - \frac{1}{2} l o g_{2} x} ⟺ \frac{1}{2} lo g_{2}^{2} x ⩾ 1 - \frac{1}{2} lo g_{2} x ⟺ (lo g_{2} x + 2) (lo g_{2} x - 1) ⩾ 0 ⟺ {x > 0 (x - 1/4) (x - 2) ⩾ 0 ⟺ x \in (0, \frac{1}{4}] \cup [2, + \infty) .

Решите неравенство $x^{l o g_{2} x} ⩾ \frac{2}{x} .$

x^{l o g_{2} x} ⩾ \frac{2}{x} ⟺ 2^{1/2 l o g_{2}^{2} x} ⩾ 2^{1 - \frac{1}{2} l o g_{2} x} ⟺ \frac{1}{2} lo g_{2}^{2} x ⩾ 1 - \frac{1}{2} lo g_{2} x ⟺ (lo g_{2} x + 2) (lo g_{2} x - 1) ⩾ 0 ⟺ {x > 0 (x - 1/4) (x - 2) ⩾ 0 ⟺ x \in (0, \frac{1}{4}] \cup [2, + \infty) .