In order to keep longer in their own storage, the supermarkets chill most of their carrots and I suspect they over do it anyway. Problem is when they are put out onto the shop floor and warm up they get damp from water vapour in the air and also actual condensation. This moisture on the surface of carrots that have been very thoroughly washed mechanically (so they look pretty of course) makes the surface cells softer and accelerates the decaying process. Can help to take them out of the bags, carefully dry each one and wrap in kitchen roll and store in a paper bag.
Buying htem loose gereally works slightly better than pre-packaged ones, but they too tend to turn mushy from the ends inwards.
By contrast, home grown or farmers carrots that are not vigorusly mechanically washed (just the dried earth carefully eased off with a thumb) Last much longer, my late stepfather used to stash them in boxes of dry sand with varying degrees of success.
Some of the supermarkets fruit is even worse for rotting quickly - not so much 'ripen at home' as 'liquefy at home'.
