It definitely works though. At 3 weeks post I usually do not have any major new growth. At 3 weeks post while eating 1-2 carrots a day and I have so much new growth it is beginning to coil up and this normally happens when I am anywhere past 8 weeks post. I can no longer even wear my bangs as they have begun to grow out and the only way to have them lay flat would be to flat iron the new growth and I'm not a fan of using lots of direct heat. I am going to continue my experiment and hopefully I will be able to stretch to 14 weeks to relax just in time for my birthday.
It's not necessarily carrots that grow your hair. It is the beta carotene. I would even go so far as to say that you could eat other foods with the orange/yellow pigment and also dark green and have the same benefits. Beta carotene is converted in the body to Vitamin A which is essential for bone and tooth development and healthy hair, skin and nails.
I would also like to add that I have a very healthy diet. I do not eat any processed foods nor any added sugars. My growth rate had increased since I began eating more peas, beans, fruits and vegetables in an effort to clear my skin. The extra beta carotene has just taken it a step further and has perhaps helped my hair to grow at its optimum rate.