Exhaustively is the adverb of exhaustive: it describes doing something in a way that covers absolutely everything. "The topic has been exhaustively researched" means researchers have gone through it completely, leaving no part unexamined. You will most often meet it beside verbs like研究-type words — researched, studied, tested, documented, examined, catalogued. It sits comfortably in academic writing, reports, and reviews. Because it praises thoroughness, it can be used to shut down an argument: saying a question has been "exhaustively studied" hints that there is little point in going over it again. Occasionally, in reviews or complaints, it tips into a mild criticism — "the report exhaustively lists every meeting" can suggest the writer went on too long. Context tells you which one is meant.