5. Given that $a, b, c$ are all positive numbers, then $\min \left\{\frac{1}{a}, \frac{2}{b}, \frac{4}{c}, \sqrt[3]{a b c}\right\}$ has the maximum value of
Consider $\sqrt{2}$.
Let $M=\min \left\{\frac{1}{a}, \frac{2}{b}, \frac{4}{c}, \sqrt[3]{a b c}\right\}$, then $M \leq \frac{1}{a}, M \leq \frac{2}{b}, M \leq \frac{4}{c}$, so $M^{3} \leq \frac{8}{a b c}$, which gives
$$
\begin{array}{c}
\sqrt[3]{a b c} \leq \frac{2}{M} \\
M \leq \sqrt[3]{a b c}
\end{array}
$$

From (1) and (2), we get $M \leq \frac{2}{M}$, so $M \leq \sqrt{2}$, i.e., $M_{\max }=\sqrt{2}$, which is achieved when $c=2 b=4 a=2 \sqrt{2}$.