OK - here's the catch. It's impossible to finish the game without collecting the carrot. Therefore, technically you can't get less than 100,000 points. My best run was this:
Low ScoreI tried to see if I could jump around the carrot, but to no avail. (which is also how I ended up with another life). I was unable to finish the game without some points, but I probably could have done a little better than this. Just a practice to see what was possible.