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:
“Society’s double behavioral standard for women and for men is, in fact, a more effective deterrent than economic discrimination because it is more insidious, less tangible. Economic disadvantages involve ascertainable amounts, but the very nature of societal value judgments makes them harder to define, their effects harder to relate.”
—Anne Tucker (b. 1945)
“Truth cannot be defined or tested by agreement with ‘the world’; for not only do truths differ for different worlds but the nature of agreement between a world apart from it is notoriously nebulous. Rather—speaking loosely and without trying to answer either Pilate’s question or Tarski’s—a version is to be taken to be true when it offends no unyielding beliefs and none of its own precepts.”
—Nelson Goodman (b. 1906)
“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)