On a joint return that question applies to both joint taxpayers so the answer is they lived with the taxpayer the whole year (lived with either one or both taxpayers on the joint return). The "you" in the question is poor wording, but it applies to either taxpayer on the joint return, only one of which needs to have lived with the children.
In addition, as Texas Roger said, if the home where you spouse and children live is your domicile (where you reside, get your mail, vote, etc) and you are only away temporary in order to work, then temporary absences count as time lived at home, unless you have established residence in another location.
**Disclaimer: This post is for discussion purposes only and is NOT tax advice. The author takes no responsibility for the accuracy of any information in this post.**
