Question

What is the difference between rolle's theorem and mean value theorem?

Answer 1

Rolle's Theorem is a special case of the Mean Value Theorem.

Explanation:

Difference 1 Rolle's theorem has 3 hypotheses (or a 3 part hypothesis), while the Mean Values Theorem has only 2.

Difference 2 The conclusions look different.

BUT
If the third hypothesis of Rolle's Theorem is true (`f(a) = f(b)`), then both theorems tell us that there is a `c` in the open interval `(a,b)` where `f'(c)=0`.

The difference really is that the proofs are simplest if we prove Rolle's Theorem first, then use it to prove the Mean Value Theorem. After we have done that, we don't really need Rolle's Teorem for anything else. I mean, every other use (other than proving Mean Value) for Rolle's Theorem can be handled by the Mean Value Theorem.