Question 24
(α) For every real number $x \gt 0$ prove that: $x^3 - 3x \ge -2$.
(β) For all real numbers $x, y, z \gt 0$, prove that:
$$
\frac{x^2 y}{z} + \frac{y^2 z}{x} + \frac{z^2 x}{y} + 2 \left( \frac{y}{xz} + \frac{z}{xy} + \frac{x}{yz} \right) \ge 9. \quad (1)
$$
When equality holds?
(β) For all real numbers $x, y, z \gt 0$, prove that:
$$
\frac{x^2 y}{z} + \frac{y^2 z}{x} + \frac{z^2 x}{y} + 2 \left( \frac{y}{xz} + \frac{z}{xy} + \frac{x}{yz} \right) \ge 9. \quad (1)
$$
When equality holds?