proof of mean value theorem

Define h(x) on [a,b] by


Clearly, h is continuousMathworldPlanetmath on [a,b], differentiableMathworldPlanetmathPlanetmath on (a,b), and


Notice that h satisfies the conditions of Rolle’s Theorem. Therefore, by Rolle’s Theorem there exists c(a,b) such that h(c)=0.
However, from the definition of h we obtain by differentiationMathworldPlanetmath that


Since h(c)=0, we therefore have


as required.


  • 1 Michael Spivak, Calculus, 3rd ed., Publish or Perish Inc., 1994.
Title proof of mean value theorem
Canonical name ProofOfMeanValueTheorem
Date of creation 2013-03-22 12:40:57
Last modified on 2013-03-22 12:40:57
Owner Andrea Ambrosio (7332)
Last modified by Andrea Ambrosio (7332)
Numerical id 5
Author Andrea Ambrosio (7332)
Entry type Proof
Classification msc 26A06