Question 179
Find the real numbers $x$, $y$, $z \gt 0$ for which
$$
xyz \le \min \left\{ 4 \left(x - \frac{1}{y}\right), 4 \left(y - \frac{1}{z}\right), 4 \left(z - \frac{1}{x}\right) \right\}.
$$
$$
xyz \le \min \left\{ 4 \left(x - \frac{1}{y}\right), 4 \left(y - \frac{1}{z}\right), 4 \left(z - \frac{1}{x}\right) \right\}.
$$