@Did The question of what kinds of tools are required in order to get certain jobs done is hard. The question there is "Do we need the derivative to show that there are exactly two solutions?"
Maybe "deep" is the wrong word, but it is certainly a much easier question to ask than to answer
Keep an eye out for this user whose recent answers have been... rather ... poor. Their response to other users's comments is defensive, and rather offensive at times.
Interestingly, @user21820, the questions date back to 2012, with nothing since; yet the user just appeared here today, 2018, two hours ago. Perhaps they created another account(s), since then.