Решите неравенство _(3/2) x^(3/2) + _(2/3) x^(2/3) (5)/(6) * x * _6 x .

Задача №17741: Неравенство

Решите неравенство $lo g_{3/2} x^{3/2} + lo g_{2/3} x^{2/3} ⩽ \frac{5}{6} \cdot x \cdot lo g_{6} x .$

lo g_{3/2} x^{3/2} + lo g_{2/3} x^{2/3} ⩽ \frac{5}{6} \cdot x \cdot lo g_{6} x ⟺ \frac{3}{2} lo g_{3/2} x - \frac{2}{3} lo g_{3/2} x ⩽ \frac{5}{6} \cdot x \cdot lo g_{6} x ⟺

⟺ lo g_{3/2} x ⩽ x \cdot lo g_{6} x ⟺ (x - lo g_{3/2} 6) lo g_{6} x ⩾ 0 ⟺

⟺ {x > 0 (x - lo g_{3/2} 6) (x - 1) ⩾ 0 ⟺ x \in (0, 1] \cup [lo g_{3/2} 6, + \infty) .

Решите неравенство $lo g_{3/2} x^{3/2} + lo g_{2/3} x^{2/3} ⩽ \frac{5}{6} \cdot x \cdot lo g_{6} x .$

lo g_{3/2} x^{3/2} + lo g_{2/3} x^{2/3} ⩽ \frac{5}{6} \cdot x \cdot lo g_{6} x ⟺ \frac{3}{2} lo g_{3/2} x - \frac{2}{3} lo g_{3/2} x ⩽ \frac{5}{6} \cdot x \cdot lo g_{6} x ⟺

⟺ lo g_{3/2} x ⩽ x \cdot lo g_{6} x ⟺ (x - lo g_{3/2} 6) lo g_{6} x ⩾ 0 ⟺

⟺ {x > 0 (x - lo g_{3/2} 6) (x - 1) ⩾ 0 ⟺ x \in (0, 1] \cup [lo g_{3/2} 6, + \infty) .