$23 \cdot 13$ If one edge length of a tetrahedron is $x$, and the other edge lengths are all 1, the volume is $F(x)$, then the function $F(x)$ on its domain.
(A) is an increasing function but has no maximum value.
(B) is an increasing function and has a maximum value.
(C) is not an increasing function and has no maximum value.
(D) is not an increasing function but has a maximum value.
(China High School Mathematics League, 1984)
[Solution] In a tetrahedron, the angle $\theta$ between two faces with three edges of length 1 each is related to $x$. The larger $\theta$ is, the larger $x$ is.
When $0^{\circ} < \theta < 90^{\circ}$, the volume $F(x)$ of the tetrahedron decreases as the angle increases, i.e., the larger $x$ is, the smaller $F(x)$ is.

In summary, $F(x)$ is not an increasing function over its domain, but it does have a maximum value.
Therefore, the correct choice is $(D)$.