If a passive construction does not make sense, you know that it at least is not transitive (However, it is not the case that all constructions that can be passivized are actually transitive. Inergative intransitives in English are also able to passivize "The building collapsed" can be rephrased as "The building was collapsed" just fine but this is also an intransitive)
So if you rephrase "He grew up in a little farm" as "In a little farm was grown up by him", we can see that the resulting sentence is ungrammatical and that, therefore, "grown up" is not a transitive verb.
If we rephrase "She picked up a card" as "A card was picked up (by her)", we can see that the result is grammatical and conclude that "picked up" could be a transitive verb. Because we have two arguments here (the thing picked up and the picker upper), we also know that it is indeed transitive and not an inergative.
I hope this clears things up. I can explain more thoroughly if you'd like.