## What is the Extended Mean Value Theorem?

There are several “extended mean value theorems”. Most authors, when referencing the EMVT are usually referring to **Cauchy’s (Extended) Mean Value Theorem** although in some cases they are referring to the mean value theorem for integrals or even Taylor’s theorem.

The “Classical” Mean Value Theorem states:

For a continuous function

gdefined on the closed interval [a, b] and differentiable on the open interval (a, b), there is at least one number c in (a, b) for which g′(c) = (g(b) – g(a)) / (b – a).

New extensions of the MVT are being developed all of the time, for a wide variety of applications including for one-sided differentiable functions [1] and holomorphic Functions [2].

## Cauchy’s Mean Value Theorem

**Cauchy’s (extended) mean value theorem** is a generalization of the classical MVT. It is usually stated as follows: If functions *f* and *g* are continuous of the closed interval [*a*, *b*] and differentiable on the open interval (*a*, *b*), then there is at least one point *c* ∈ (*a*, *b*) such that:

As long as g′ (x) ≠ 0.

Geometrically, the theorem is telling us that there is a value c in the open interval (a, b) for which the tangent line to the curve at (f(c), g(c)) is parallel to the line that connects the two endpoints:

## Other Extended Mean Value Theorem / Special Cases

**Rolle’s theorem**: A special case of the MVT, when f(a) = f(b)- The
**mean value theorem for integrals**: states that somewhere under the curve of a function, there is a rectangle with an area equal to the whole area under a curve. **Taylor’s Theorem**: Although some authors refer to this as an extension of the MVT [3], it could be viewed more like an application. The MVT can be used to prove the a generalized Taylor’s theorem (with Lagrange form of the remainder term) [4] or deduce Taylor’s theorem in one variable [5].

## References

[1] Slota, D. et al. (2012). Mean Value Theorems for One-Sided Differentiable Functions. PDF: http://www.math.put.poznan.pl/artykuly/FM48(2012)-WitulaR-HetmaniokE-SlotaD.pdf

[2] Cakmak, D. & Tiryaki, A. Mean Value Theorem for Holomorphic Functions. Electronic Journal of Differential Equations, Vol. 2012 (2012), No. 34, pp. 1–6.

ISSN: 1072-6691Retrieved May 6, 2021 from: https://ejde.math.txstate.edu/Volumes/2012/34/cakmak.pdf

[3] Robbin J. (2000). Taylor’s Formula (The Extended Mean Value Theorem). Retrieved May 6, 2021 from: https://people.math.wisc.edu/~robbin/221dir/taylor.pdf

[4] Yunsen, C. Mean value theorem. Retrieved May 6, 2021 from: https://www.academia.edu/23708876/Mean_value_theorem

[5] Pinkham, H. (2014). Analysis, Convexity, and Optimization. Lecture 4: Taylor’s Theorem in One Variable. Retrieved May 6, 2021 from: https://www.math.columbia.edu/~pinkham/Optimizationbook.pdf

**CITE THIS AS:**

**Stephanie Glen**. "Extended Mean Value Theorem" From

**CalculusHowTo.com**: Calculus for the rest of us! https://www.calculushowto.com/extended-mean-value-theorem/

**Need help with a homework or test question? **With Chegg Study, you can get step-by-step solutions to your questions from an expert in the field. Your first 30 minutes with a Chegg tutor is free!