Suppose $f(t) \ge 0$ for $t\ge 0$, is it the case that $$\int_0^\infty f(t) \sin(x t) dt \ge 0$$ if and only if $$\frac{d}{dt}f(t) \le 0\ \ ?$$
One direction is fairly trivial involving an alternating series. The other direction is not so obvious, if it is even true. Does anyone have a proof or a reference?