  Let $n$ and $k$ be nonnegative integers such that $1\le k\le n+1$. The convex
cone $\mathcal{F}_+^{k:n}$ of all functions $f$ on an arbitrary interval
$I\subseteq\mathbb{R}$ whose derivatives $f^{(j)}$ of orders $j=k-1,\dots,n$
are nondecreasing is characterized in terms of extreme rays of the cone
$\mathcal{F}_+^{k:n}$. A simple description of the convex cone dual to
$\mathcal{F}_+^{k:n}$ is given. These results are useful in, and were motivated
by, applications in probability. In fact, the results are obtained in a more
general setting with certain generalized derivatives of $f$ of the $j$th order
in place of $f^{(j)}$. Somewhat similar results were previously obtained in the
case when the left endpoint of the interval $I$ is finite, with certain
additional integrability conditions; such conditions fail to hold in the
mentioned applications.
