proof of extended mean-value theorem

Let f:[a,b] and g:[a,b] be continuousMathworldPlanetmath on [a,b] and differentiableMathworldPlanetmathPlanetmath on (a,b). Define the function


Because f and g are continuous on [a,b] and differentiable on (a,b), so is h. Furthermore, h(a)=h(b)=0, so by Rolle’s theorem there exists a ξ(a,b) such that h(ξ)=0. This implies that


and, if g(b)g(a),

Title proof of extended mean-value theorem
Canonical name ProofOfExtendedMeanvalueTheorem
Date of creation 2013-03-22 13:09:00
Last modified on 2013-03-22 13:09:00
Owner pbruin (1001)
Last modified by pbruin (1001)
Numerical id 6
Author pbruin (1001)
Entry type Proof
Classification msc 26A06
Synonym proof of Cauchy’s mean-value theorem
Related topic MeanValueTheorem