Health department food safety

The comment of local food safety is spot on, they are the authority on legality for you and that is one of the criteria you must meet for your own protection and that of your clients/guests. As you ask specifically about health department requirements, they are the authority, so go straight to the source to satisfy them and that is more of a legal question, so outside of what we can really answer.

Bacterial growth danger zone is 40-140F, so in general the advice is usually to stay out of that zone. Now, on you specific item, raw carrots are not going to be a high illness risk I would think. However, going much into that range is also going to put you into a zone that the quality, not just the safety of the product will deteriorate more quickly if you are in that range. Personal observation is that carrot will hold up fairly well up into the 50's, but why? If you are in a salad bar setup, you likely have other items that may deteriorate more quickly so to handle them you want temps at least close to that 40F number or lower such as using an ice bath if the items will remain for an extended time. And if you have items such as those made with mayonnaise, boiled eggs or other cooked items, any meats, those must be outside that high growth zone so above 140F for hot items and below 40F or so for cold. If you need to chill or keep warm some items, you might as well do all.