#According to my textbook:
Rolle's theorem states that if a function $f$ is continuous on the closed interval $[a, b]$ and differentiable on the open interval $(a, b)$ such that $f(a) = f(b)$, then $f′(x) = 0$ for some $x$ with $a ≤ x ≤ b$.
#According to Wikipedia:
If a real-valued function $f$ is continuous on a proper closed interval $[a, b]$, differentiable on the open interval $(a, b)$, and $f(a) = f(b)$, then there exists at least one $c$ in the open interval (a, b) such that
$f'(c)=0$.
So one definition says that $c$ should belong in closed interval $[a,b]$ but the other says that $c$ should be in open interval $(a,b)$.
Which definition is correct ? Why?
Answer
These are theorems, not definitions, and both of them are correct. Notice that if Wikipedia is correct, then your textbook is automatically correct as well: if there exists $c\in (a,b)$ such that $f'(c)=0$, then there also exists $x\in [a,b]$ such that $f'(x)=0$, since you can take $x=c$ (since $(a,b)$ is a subset of $[a,b]$). On the other hand, you can't (in any obvious way) deduce Wikipedia's statement from your textbook's, so Wikipedia's statement is stronger: it tells you more information. So you could say Wikipedia's statement is more useful or more powerful, and is "correct" in that you might as well use it instead of your textbook's version.
As for which one is "correct" in the sense of being the "standard" statement of Rolle's theorem, I would say the Wikipedia version is probably more standard. But mathematical theorems quite often do not have universally accepted "standard" versions and instead have several different versions that are closely related but may be slightly different and all tend to be referred to with the same name. It's not like there's some committee of mathematicians who gets together and declares "this is the statement we will call Rolle's theorem"; everyone just refers to theorems independently and so there ends up being some minor variation.
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