Cauchy Mean Value Theorem
For continuous on and differentiable on , there is with .
Proposition 3.15 (Cauchy Mean Value Theorem)
Let be continuous on and differentiable on . Then such that
Proof
Reduce to Rolle’s Theorem. Let
so that
Then , and Rolle’s Theorem gives with , which is the claimed identity. Taking to be the identity recovers the ordinary Mean Value Theorem.
Related
Stated in
- Proposition 3.15 (Cauchy Mean Value Theorem)§3.2 Mean Value Theorems