Mean value theorem

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 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 — Use the open-interval derivative.
• Point values and continuity of real functions — Require continuity on the closed interval.
• Average rate of change between distinct inputs — The endpoint quotient is the secant slope.
• Usual order on the real numbers — 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.