Question: It is very tempting to assign extra credit problems that are (a) substantially more difficult than others on the problem set, (b) ask for the history of something we discussed in class, or (c) are related to something just barely outside of the scope of the class.

However, I have never done so because, in my experience, very few professors give extra credit.

* **Why do so few professors assign extra credit (or is this not the case)?**
* **What are the arguments against assigning extra credit problems?**

Answer: 
**Extra credit often leads to problems with curves.** I've heard dialogues like this:

**Student:** *I got 88% overall, I think I deserve an A.*

**Teacher:** *But lots of other students got 90's by answering extra credit.*

Here the grade without extra credit is being compared against a standard of grades with extra credit. So the credit was not "extra" at all, but just a badly-labeled part of the class.

To avoid this, keep two sets of grade books, and don't look at the one with extra credit until you've set grade cutoffs from the one without extra credit.

Or, more easily and more transparently, don't give extra credit at all. Your suggestions a, b, and c sound good for short required written assignments.
I give extra credits routinely, mostly to reward not-asked-for work (point out an error in the class notes, suggest an interesting alternative solution for an exam or homework problem, ...) and sometimes for extra homework assigned. In my (by now extensive) experience, students who go the mile to get extra credit would get good or outstanding grades even without the extra points. A very few of those hanging in the balance get over the bump by this, but then they have shown enough insight to merit a passing grade.

In exams I asign a few extra points, but not a problem marked specially. That gives some leeway for mistakes and still getting a perfect score. My exams are typically a fairly simple, general question ("do you even know what we are talking about?") and the remainder several more difficult problems covering the material about evenly. Sometimes questions have a "bonus track" part, if it makes sense to look somewhat further.