Mean Value Theorem
If is continuous on and differentiable on , then such that .
Theorem 3.9 (Mean Value Theorem)
Let be continuous on and differentiable on . Then such that
Reformulation
Given with , the theorem says there is such that
Unlike the affine characterization of differentiability, this statement carries no error term: for a real function on an interval, the increment is exactly a step of the derivative at an intermediate point.
Proof
Subtract the line through and by considering
where is the chord. Then and . By Rolle’s Theorem there is with , that is .
Related
Stated in
- Theorem 3.9 (Mean Value Theorem)§3.2 Mean Value Theorems