Infinitesimal calculus MOC

Rolle’s theorem

Let be continuous with . If for every the limits and have , then there exists some such that contains both a positive and negative (but possibly infinite) number. anal

Proof

By the extreme value theorem reaches either its maximum or minimum at , for if both lay on the boundary then would be constant. Without loss of generality assume has a maximum at (otherwise consider ). Clearly

and similarly

as required.

In case is differentiable, Rolle’s theorem is equivalent to saying there exists some such that , which itself is a special case of the mean value theorem.

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