proof of mean value theorem


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

h(x)=f(x)-f(a)-(f(b)-f(a)b-a)(x-a)

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

h(a)=f(a)-f(a)=0h(b)=f(b)-f(a)-(f(b)-f(a)b-a)(b-a)=0

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

h(x)=f(x)-f(b)-f(a)b-a

Since h(c)=0, we therefore have

f(c)=f(b)-f(a)b-a

as required.

References

  • 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