The mean-value point of a quadratic

Goal and scope

Directly construct a mean-value point for f(t)=A*t^2+B*t+C on a<b.

An exact polynomial family, not a sampling argument for general continuity or differentiability.

Givens and symbols

• algebra
• polynomial

• G1 — A,B,C,a,b are rational, a<b, and f(t)=A*t^2+B*t+C.

• c — strict-interior mean-value witness
• A,B,C — quadratic, linear, constant coefficients

Rules and sources

• mko_definition · cited — 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. · mko-mean-value-theorem
• algebra · prerequisite — Use real-field identities; division requires a nonzero denominator and multiplication by a positive number preserves inequalities.
• polynomial · prerequisite — Real polynomials are continuous and differentiate termwise: the derivative of x^k is k*x^(k-1), and constants differentiate to zero.

Main derivation

f(t)=At^2+Bt+C

f\text{ continuous on }[a,b]\text{ and differentiable on }(a,b)

f(b)-f(a)

A(b^2-a^2)+B(b-a)=(b-a)(A(a+b)+B)

(f(b)-f(a))/(b-a)

A(a+b)+B

f(t)=At^2+Bt+C

f'(t)=2At+B

c=(a+b)/2

c-a=(b-a)/2>0,\quad b-c=(b-a)/2>0

f'(c)=2A(a+b)/2+B

f'(c)=A(a+b)+B=(f(b)-f(a))/(b-a)

2Ac+B=A(a+b)+B

A\ne0\Longrightarrow c=(a+b)/2;\quad A=0\Longrightarrow\forall c\in(a,b)\text{ works}

A=2,B=-3,C=1,a=0,b=2

f(0)=1,\ f(2)=3,\ c=1,\ f'(1)=1=(3-1)/2

Substeps and lemmas

No substeps in this guide.

Reverse audits

• S08 · premise_audit · S06 — The polynomial family was proved earlier; this numerical check is only an audit.
• S07 · premise_audit · S03, S04 — Existence does not imply uniqueness, and division by A is forbidden in the zero branch.
• S06 · premise_audit · S03, S04, S05 — Equality and interior membership were separately checked; a numerical equality is not called a general MVT proof.
• S05 · premise_audit · G1 — A mean-value witness cannot merely be an endpoint.
• S04 · premise_audit · G1 — This is not only a numerical derivative at a candidate c.
• S03 · inverse_equivalence · G1, S02 — This division is unavailable when a=b.
• S02 · inverse_equivalence · G1 — A/B/C remain quadratic, linear and constant coefficients in that order.
• S01 · premise_audit · G1 — Closed-interval continuity and open-interval differentiability are distinct obligations.

Error controls and negative controls

Denominator first

Verify a<b before dividing by b-a.

Split interval premises

The full polynomial supplies continuity on the whole closed interval and differentiability on the whole interior.

• The secant division remains valid for a=b. → reject — The denominator is zero.
• MVT applies directly to |t| on [-1,1]. → reject — It is not differentiable at zero, and no interior derivative equals the secant slope zero.
• The mean-value point must be unique. → reject — When A=0 every interior point works.

Result and limitations

The midpoint is always a mean-value point for this quadratic family; it is unique for A≠0, while every interior point works for A=0.

Verification and source binding

1 declared checks for this guide; only the listed expressions are covered.

• source commit: b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f
• adopted source SHA-256: e1932f3808c6a6878c0dea094cd386502a613adef2e578fbef51aa5a8ba53736
• original source SHA-256: e1932f3808c6a6878c0dea094cd386502a613adef2e578fbef51aa5a8ba53736
• Lean: NOT_RUN · independent AI: SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW

Same-source full JSON (formulas, ASTs, controls and revision log)