Standard Version of The Theorem
If a real-valued function ƒ is continuous on a closed interval, differentiable on the open interval (a, b), and ƒ(a) = ƒ(b), then there exists a c in the open interval (a, b) such that
This version of Rolle's theorem is used to prove the mean value theorem, of which Rolle's theorem is indeed a special case. It is also the basis for the proof of the Taylor's theorem.
Read more about this topic: Rolle's Theorem
Famous quotes containing the words standard, version and/or theorem:
“Error is a supposition that pleasure and pain, that intelligence, substance, life, are existent in matter. Error is neither Mind nor one of Mind’s faculties. Error is the contradiction of Truth. Error is a belief without understanding. Error is unreal because untrue. It is that which seemeth to be and is not. If error were true, its truth would be error, and we should have a self-evident absurdity—namely, erroneous truth. Thus we should continue to lose the standard of Truth.”
—Mary Baker Eddy (1821–1910)
“If the only new thing we have to offer is an improved version of the past, then today can only be inferior to yesterday. Hypnotised by images of the past, we risk losing all capacity for creative change.”
—Robert Hewison (b. 1943)
“To insure the adoration of a theorem for any length of time, faith is not enough, a police force is needed as well.”
—Albert Camus (1913–1960)