Assuming there is one carrot: In a duel going after the carrot right after you make the score isn't always the best idea, in fact very often it's a good idea to avoid it even if you are at 1. This is counterintuitive at first but makes sense. This is because your opponent knows you are hurt, whereas you know your opponent is at 3 because they just spawned. If they camp the carrot, you will be an easy kill assuming you go right to it after making the score. But if you ignore the carrot entirely and they camp it, you can go grab the enemy flag and all the powerups. Once you get their flag, they will probably go try to get yours, which opens up the carrot for you to easily take (they couldn't have taken it because they are at 3).

In other words, I would argue that after making a kill in a ctf duel, it's more often better to score, then capture the enemy flag, THEN get the carrot. This does not apply to all levels and all situations though. For example none of this applies if there are multiple carrots, and it's good to switch it up once and a while so you are unpredictable.