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Extended Mean Value Theorem

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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 g defined 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:
cauchy's extended mean value theorem

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:
cauchy extended mean value theorem graphic

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


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Stephanie Glen. "Extended Mean Value Theorem" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/extended-mean-value-theorem/
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