1.) "live" is clearly a canonical form

(Wehmeyer; Ashby 2000, p.752 f.)

2.) "furiously" is an adverb: adverbs are often entried "run on" to adjective forms

(Wehmeyer; Ashby 2000, p.523)

3.) "purposive" is an adjective which is derived from the word "purpose", some dictionaries might list it "run on" but the OALD decided to list is as entry term because this dictionary is made for foreign speakers which need more guidance

4.) "on purpose" consists of a preposition and the entry term "purpose", it does not deserve an own entry and is a form which depends on the entry term itself

(Wehmeyer; Ashby 2000, p.1029)