I have absolutely no idea how to show this:
Let $X$ be a random variable whose distribution is not degenerate. By considering the function $F( \theta) = \mathbb{E} U( \theta X)$, $\theta \in \mathbb{R}$, where $U: \mathbb{R} \rightarrow (-\infty, 0)$ is strictly increasing, concave and continuously differentiable, show that
EITHER there exists some positive integrable $Z$ such that $\mathbb{E} (XZ) =0 \,$
OR one of $X, -X$ is almost surely nonnegative.
(Note the facts that $F$ is concave and is continuously differentiable.)