Generalization To Higher Derivatives
We can also generalize Rolle's theorem by requiring that f has more points with equal values and greater regularity. Specifically, suppose that
- the function f is n − 1 times continuously differentiable on the closed interval and the nth derivative exists on the open interval (a,b), and
- there are n intervals given by a1 < b1 ≤ a2 < b2 ≤ . . .≤ an < bn in such that f(ak) = f(bk) for every k from 1 to n.
Then there is a number c in (a,b) such that the nth derivative of f at c is zero.
Of course, the requirements concerning the nth derivative of f can be weakened as in the generalization above, giving the corresponding (possibly weaker) assertions for the right- and left-hand limits defined above with f (n−1) in place of f.
Read more about this topic: Rolle's Theorem
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