16.1. (CSSR, 79). Cubes are considered whose centers coincide with the center of symmetry of a given rectangular parallelepiped with edges $a<b<c$, and whose faces are parallel to those of the latter. Find the edge of the cube that has the smallest difference between the volumes of the union and intersection with this parallelepiped.
16.1. If we denote the edge of the cube by $x$, then the difference in volumes specified in the problem is

$$
f(x)=\left\{\begin{array}{lc}
a b c-x^{3} & \text { for } \quad 00$, and its derivative is

$$
f^{\prime}(x)= \begin{cases}-3 x^{2} & \text { for } 03 b^{2}-2 a b>0$ ), and on the interval ( $a$; b) it either decreases if $b \leqslant 4 a / 3$ (since $3 x^{2}-4 a x4 a / 3$. Thus, the minimum value of the function $f(x)$ is either at $x=b$ (if $b \leqslant 4 a / 3$), or at $x=4 a / 3$ (if $b>4 a / 3$), and the sought value of $x$ is $\min \{b ; 4 a / 3\}$.