completeness of a vector field fX, with X complete and f>0

Let $M$ be a noncompact $C^\infty$ manifold, let $X$ be a complete $C^\infty$ vector field on $M$, and take $f\in C^\infty\big(M;(0,\infty)\big)$ a strictly positive function.

Question: Does anyone know sufficient conditions on the function $f$ implying the completeness of the vector field $fX$ ?

(When $M$ is compact, the vector field $fX$ is complete and has the same integral curves as the vector field $X$, cf. Chapter 2, Section 2 of the book of Ergodic Theory of Cornfeld, Fomin and Sinai.)

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If $f$ is bounded, then the rate at which one travels along the integral curves of $X$ is only increased by a bounded factor, so it still takes infinite time to get all the way along each integral curve,so $fX$ is complete.
The first step in a proof is to notice that the vector field $X$ restricts to a vector field along each of its integral curves, so we only need to work on one integral curve at a time. So without loss of generality, our manifold is one dimensional. It is also easy to pull back to the universal covering space, because completeness of vector fields is preserved and reflected by covering maps. You can then assume that your manifold is the real number line. You can use the flow of $X$ to parameterize the real number line, so then $X=d/dt$. After that the proof is elementary. – Ben McKay Feb 21 2012 at 9:14