# Mean value theorem

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

If a<b, f is continuous on [a,b] and differentiable on (a,b), some interior c has derivative equal to the endpoint secant slope.

mko-mean-value-theorem · theorem · v0.10.0



## Mathematical statement

Question parameters A/B/C explicitly encode f(t)=A*t^2+B*t+C. Require a<b, continuity on the closed interval and differentiability on the open interval. The witness satisfies a<c<b and f'(c)=(f(b)−f(a))/(b−a). It need not be unique: affine and constant cases can have many witnesses. Finite samples, a=b, a jump or an interior cusp do not supply the required theorem hypotheses.





## Representative formula (exact TeX)

D(f,c)=\frac{f(b)-f(a)}{b-a}





## Assumptions


- Positive families declare the integer-coefficient polynomial f(t)=A*t^2+B*t+C, so global continuity and differentiability hold.
- Endpoints and witnesses are checked with exact integer/rational arithmetic.





## Symbol scope


- D — derivative evaluation
- f — real function
- c — interior witness
- a — left endpoint
- b — right endpoint





## Intuition, definition, and misconceptions

A/B/C are the quadratic, linear and constant coefficients in order.

Continuity and differentiability concern different parts of the interval.

a<b makes the secant denominator nonzero; witness existence does not imply uniqueness.

Finite samples that look smooth do not establish differentiability; c may not be an endpoint.





## Adopted derivation guide

[The mean-value point of a quadratic](https://ocme.evemisslab.com/en/derivations/derivation-mko-mean-value-theorem-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Real derivatives at a point and finite difference quotients](https://ocme.evemisslab.com/en/mko/mko-derivative/) — Use the open-interval derivative.
- [Point values and continuity of real functions](https://ocme.evemisslab.com/en/mko/mko-continuity/) — Require continuity on the closed interval.
- [Average rate of change between distinct inputs](https://ocme.evemisslab.com/en/mko/mko-rate-of-change/) — The endpoint quotient is the secant slope.
- [Usual order on the real numbers](https://ocme.evemisslab.com/en/mko/mko-order-relation/) — Endpoints are ordered and the witness is interior.





## Verification boundary

Exact families cover declared polynomials and named boundary cases, not a sampling proof of general differentiability.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.





## Next obligation

Add Taylor polynomials and remainders.





## Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.

[Canonical MKO JSON](https://ocme.evemisslab.com/data/mko/mean-value-theorem.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-mean-value-theorem/index.md)