Exhaustive means performed comprehensively and completely. When you recruit a new employee (or spouse), you undertake an exhaustive search for the best talent.
When you are exhaustive about something, you are testing all possibilities or considering all elements. If you want to become an attorney you will need an exhaustive knowledge of the leather bound books in the law library. When you exhaust something, you use it up entirely, so something exhaustive is complete. After your exhaustive tour of Rome, you're exhausted.