Question 133
Find the smallest value of
$$
xy^3z^2 + \frac{4z}{x} - 8yz - \frac{4}{xy}
$$
where $x$, $y$, $z$ are positive real numbers satisfying at least one of the following inequalities:
$$
2xy \gt 1 \text{ and } yz \gt 1.
$$
$$
xy^3z^2 + \frac{4z}{x} - 8yz - \frac{4}{xy}
$$
where $x$, $y$, $z$ are positive real numbers satisfying at least one of the following inequalities:
$$
2xy \gt 1 \text{ and } yz \gt 1.
$$