# Exact Polynomial Secants, Interior MVT Witnesses and Mode-Specific Boundaries

This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed.

This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.

batch-20261014-advance-20261110-r02-mean-value-theorem-foundations · 2026-10-14 · 100 questions

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

Field order uses a source-bound presentation supplement; canonical JSONL and mathematical answers are unchanged.

[Field-order and source-binding data](https://ocme.evemisslab.com/data/i18n/question-output-protocols.json)



## 1. Continuity and Differentiability Apply on Different Interval Parts

Use the whole real polynomial f(t)=A*t^2+B*t+C with A=1, B=-10, C=1, a=-2, b=3 and a<b. Coefficients are quadratic, linear and constant in that order. This declared polynomial is continuous on [a,b] and differentiable on (a,b); finite samples do not certify those hypotheses for an arbitrary function. Give (a, b, ordered_endpoints, closed_interval_continuity, open_interval_differentiability, certificate).

View answer and explanation(-2, 3, Yes, Yes, Yes, polynomial_continuous_on_closed_interval_and_differentiable_on_open_interval)

Audit a<b, continuity on the closed interval and differentiability on the open interval. A theorem witness must lie strictly inside (a,b); existence does not imply uniqueness. The exact result, retaining the requested order, is (-2, 3, Yes, Yes, Yes, polynomial_continuous_on_closed_interval_and_differentiable_on_open_interval).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r02-mean-value-theorem-0001/)

Mean value theorem · qf-mod41-mvt-hypothesis-audit · candidate





## 2. Continuity and Differentiability Apply on Different Interval Parts

Use the whole real polynomial f(t)=A*t^2+B*t+C with A=2, B=-9, C=2, a=-3, b=4 and a<b. Coefficients are quadratic, linear and constant in that order. This declared polynomial is continuous on [a,b] and differentiable on (a,b); finite samples do not certify those hypotheses for an arbitrary function. Give (a, b, ordered_endpoints, closed_interval_continuity, open_interval_differentiability, certificate).

View answer and explanation(-3, 4, Yes, Yes, Yes, polynomial_continuous_on_closed_interval_and_differentiable_on_open_interval)

Audit a<b, continuity on the closed interval and differentiability on the open interval. A theorem witness must lie strictly inside (a,b); existence does not imply uniqueness. The exact result, retaining the requested order, is (-3, 4, Yes, Yes, Yes, polynomial_continuous_on_closed_interval_and_differentiable_on_open_interval).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r02-mean-value-theorem-0002/)

Mean value theorem · qf-mod41-mvt-hypothesis-audit · candidate





## 3. Continuity and Differentiability Apply on Different Interval Parts

Use the whole real polynomial f(t)=A*t^2+B*t+C with A=3, B=-8, C=5, a=-4, b=5 and a<b. Coefficients are quadratic, linear and constant in that order. This declared polynomial is continuous on [a,b] and differentiable on (a,b); finite samples do not certify those hypotheses for an arbitrary function. Give (a, b, ordered_endpoints, closed_interval_continuity, open_interval_differentiability, certificate).

View answer and explanation(-4, 5, Yes, Yes, Yes, polynomial_continuous_on_closed_interval_and_differentiable_on_open_interval)

Audit a<b, continuity on the closed interval and differentiability on the open interval. A theorem witness must lie strictly inside (a,b); existence does not imply uniqueness. The exact result, retaining the requested order, is (-4, 5, Yes, Yes, Yes, polynomial_continuous_on_closed_interval_and_differentiable_on_open_interval).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r02-mean-value-theorem-0003/)

Mean value theorem · qf-mod41-mvt-hypothesis-audit · candidate





## 4. Continuity and Differentiability Apply on Different Interval Parts

Use the whole real polynomial f(t)=A*t^2+B*t+C with A=4, B=-7, C=10, a=-5, b=6 and a<b. Coefficients are quadratic, linear and constant in that order. This declared polynomial is continuous on [a,b] and differentiable on (a,b); finite samples do not certify those hypotheses for an arbitrary function. Give (a, b, ordered_endpoints, closed_interval_continuity, open_interval_differentiability, certificate).

View answer and explanation(-5, 6, Yes, Yes, Yes, polynomial_continuous_on_closed_interval_and_differentiable_on_open_interval)

Audit a<b, continuity on the closed interval and differentiability on the open interval. A theorem witness must lie strictly inside (a,b); existence does not imply uniqueness. The exact result, retaining the requested order, is (-5, 6, Yes, Yes, Yes, polynomial_continuous_on_closed_interval_and_differentiable_on_open_interval).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r02-mean-value-theorem-0004/)

Mean value theorem · qf-mod41-mvt-hypothesis-audit · candidate





## 5. Continuity and Differentiability Apply on Different Interval Parts

Use the whole real polynomial f(t)=A*t^2+B*t+C with A=5, B=-6, C=17, a=-6, b=7 and a<b. Coefficients are quadratic, linear and constant in that order. This declared polynomial is continuous on [a,b] and differentiable on (a,b); finite samples do not certify those hypotheses for an arbitrary function. Give (a, b, ordered_endpoints, closed_interval_continuity, open_interval_differentiability, certificate).

View answer and explanation(-6, 7, Yes, Yes, Yes, polynomial_continuous_on_closed_interval_and_differentiable_on_open_interval)

Audit a<b, continuity on the closed interval and differentiability on the open interval. A theorem witness must lie strictly inside (a,b); existence does not imply uniqueness. The exact result, retaining the requested order, is (-6, 7, Yes, Yes, Yes, polynomial_continuous_on_closed_interval_and_differentiable_on_open_interval).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r02-mean-value-theorem-0005/)

Mean value theorem · qf-mod41-mvt-hypothesis-audit · candidate





## 6. Continuity and Differentiability Apply on Different Interval Parts

Use the whole real polynomial f(t)=A*t^2+B*t+C with A=1, B=-5, C=26, a=-7, b=8 and a<b. Coefficients are quadratic, linear and constant in that order. This declared polynomial is continuous on [a,b] and differentiable on (a,b); finite samples do not certify those hypotheses for an arbitrary function. Give (a, b, ordered_endpoints, closed_interval_continuity, open_interval_differentiability, certificate).

View answer and explanation(-7, 8, Yes, Yes, Yes, polynomial_continuous_on_closed_interval_and_differentiable_on_open_interval)

Audit a<b, continuity on the closed interval and differentiability on the open interval. A theorem witness must lie strictly inside (a,b); existence does not imply uniqueness. The exact result, retaining the requested order, is (-7, 8, Yes, Yes, Yes, polynomial_continuous_on_closed_interval_and_differentiable_on_open_interval).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r02-mean-value-theorem-0006/)

Mean value theorem · qf-mod41-mvt-hypothesis-audit · candidate





## 7. Continuity and Differentiability Apply on Different Interval Parts

Use the whole real polynomial f(t)=A*t^2+B*t+C with A=2, B=-4, C=37, a=-8, b=9 and a<b. Coefficients are quadratic, linear and constant in that order. This declared polynomial is continuous on [a,b] and differentiable on (a,b); finite samples do not certify those hypotheses for an arbitrary function. Give (a, b, ordered_endpoints, closed_interval_continuity, open_interval_differentiability, certificate).

View answer and explanation(-8, 9, Yes, Yes, Yes, polynomial_continuous_on_closed_interval_and_differentiable_on_open_interval)

Audit a<b, continuity on the closed interval and differentiability on the open interval. A theorem witness must lie strictly inside (a,b); existence does not imply uniqueness. The exact result, retaining the requested order, is (-8, 9, Yes, Yes, Yes, polynomial_continuous_on_closed_interval_and_differentiable_on_open_interval).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r02-mean-value-theorem-0007/)

Mean value theorem · qf-mod41-mvt-hypothesis-audit · candidate





## 8. Continuity and Differentiability Apply on Different Interval Parts

Use the whole real polynomial f(t)=A*t^2+B*t+C with A=3, B=-3, C=50, a=-9, b=10 and a<b. Coefficients are quadratic, linear and constant in that order. This declared polynomial is continuous on [a,b] and differentiable on (a,b); finite samples do not certify those hypotheses for an arbitrary function. Give (a, b, ordered_endpoints, closed_interval_continuity, open_interval_differentiability, certificate).

View answer and explanation(-9, 10, Yes, Yes, Yes, polynomial_continuous_on_closed_interval_and_differentiable_on_open_interval)

Audit a<b, continuity on the closed interval and differentiability on the open interval. A theorem witness must lie strictly inside (a,b); existence does not imply uniqueness. The exact result, retaining the requested order, is (-9, 10, Yes, Yes, Yes, polynomial_continuous_on_closed_interval_and_differentiable_on_open_interval).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r02-mean-value-theorem-0008/)

Mean value theorem · qf-mod41-mvt-hypothesis-audit · candidate





## 9. Continuity and Differentiability Apply on Different Interval Parts

Use the whole real polynomial f(t)=A*t^2+B*t+C with A=4, B=-2, C=65, a=-10, b=11 and a<b. Coefficients are quadratic, linear and constant in that order. This declared polynomial is continuous on [a,b] and differentiable on (a,b); finite samples do not certify those hypotheses for an arbitrary function. Give (a, b, ordered_endpoints, closed_interval_continuity, open_interval_differentiability, certificate).

View answer and explanation(-10, 11, Yes, Yes, Yes, polynomial_continuous_on_closed_interval_and_differentiable_on_open_interval)

Audit a<b, continuity on the closed interval and differentiability on the open interval. A theorem witness must lie strictly inside (a,b); existence does not imply uniqueness. The exact result, retaining the requested order, is (-10, 11, Yes, Yes, Yes, polynomial_continuous_on_closed_interval_and_differentiable_on_open_interval).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r02-mean-value-theorem-0009/)

Mean value theorem · qf-mod41-mvt-hypothesis-audit · candidate





## 10. Continuity and Differentiability Apply on Different Interval Parts

Use the whole real polynomial f(t)=A*t^2+B*t+C with A=5, B=-1, C=82, a=-11, b=12 and a<b. Coefficients are quadratic, linear and constant in that order. This declared polynomial is continuous on [a,b] and differentiable on (a,b); finite samples do not certify those hypotheses for an arbitrary function. Give (a, b, ordered_endpoints, closed_interval_continuity, open_interval_differentiability, certificate).

View answer and explanation(-11, 12, Yes, Yes, Yes, polynomial_continuous_on_closed_interval_and_differentiable_on_open_interval)

Audit a<b, continuity on the closed interval and differentiability on the open interval. A theorem witness must lie strictly inside (a,b); existence does not imply uniqueness. The exact result, retaining the requested order, is (-11, 12, Yes, Yes, Yes, polynomial_continuous_on_closed_interval_and_differentiable_on_open_interval).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r02-mean-value-theorem-0010/)

Mean value theorem · qf-mod41-mvt-hypothesis-audit · candidate





## 11. Continuity and Differentiability Apply on Different Interval Parts

Use the whole real polynomial f(t)=A*t^2+B*t+C with A=1, B=0, C=101, a=-12, b=13 and a<b. Coefficients are quadratic, linear and constant in that order. This declared polynomial is continuous on [a,b] and differentiable on (a,b); finite samples do not certify those hypotheses for an arbitrary function. Give (a, b, ordered_endpoints, closed_interval_continuity, open_interval_differentiability, certificate).

View answer and explanation(-12, 13, Yes, Yes, Yes, polynomial_continuous_on_closed_interval_and_differentiable_on_open_interval)

Audit a<b, continuity on the closed interval and differentiability on the open interval. A theorem witness must lie strictly inside (a,b); existence does not imply uniqueness. The exact result, retaining the requested order, is (-12, 13, Yes, Yes, Yes, polynomial_continuous_on_closed_interval_and_differentiable_on_open_interval).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r02-mean-value-theorem-0011/)

Mean value theorem · qf-mod41-mvt-hypothesis-audit · candidate





## 12. Continuity and Differentiability Apply on Different Interval Parts

Use the whole real polynomial f(t)=A*t^2+B*t+C with A=2, B=1, C=122, a=-13, b=14 and a<b. Coefficients are quadratic, linear and constant in that order. This declared polynomial is continuous on [a,b] and differentiable on (a,b); finite samples do not certify those hypotheses for an arbitrary function. Give (a, b, ordered_endpoints, closed_interval_continuity, open_interval_differentiability, certificate).

View answer and explanation(-13, 14, Yes, Yes, Yes, polynomial_continuous_on_closed_interval_and_differentiable_on_open_interval)

Audit a<b, continuity on the closed interval and differentiability on the open interval. A theorem witness must lie strictly inside (a,b); existence does not imply uniqueness. The exact result, retaining the requested order, is (-13, 14, Yes, Yes, Yes, polynomial_continuous_on_closed_interval_and_differentiable_on_open_interval).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r02-mean-value-theorem-0012/)

Mean value theorem · qf-mod41-mvt-hypothesis-audit · candidate





## 13. Continuity and Differentiability Apply on Different Interval Parts

Use the whole real polynomial f(t)=A*t^2+B*t+C with A=3, B=2, C=145, a=-14, b=15 and a<b. Coefficients are quadratic, linear and constant in that order. This declared polynomial is continuous on [a,b] and differentiable on (a,b); finite samples do not certify those hypotheses for an arbitrary function. Give (a, b, ordered_endpoints, closed_interval_continuity, open_interval_differentiability, certificate).

View answer and explanation(-14, 15, Yes, Yes, Yes, polynomial_continuous_on_closed_interval_and_differentiable_on_open_interval)

Audit a<b, continuity on the closed interval and differentiability on the open interval. A theorem witness must lie strictly inside (a,b); existence does not imply uniqueness. The exact result, retaining the requested order, is (-14, 15, Yes, Yes, Yes, polynomial_continuous_on_closed_interval_and_differentiable_on_open_interval).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r02-mean-value-theorem-0013/)

Mean value theorem · qf-mod41-mvt-hypothesis-audit · candidate





## 14. Continuity and Differentiability Apply on Different Interval Parts

Use the whole real polynomial f(t)=A*t^2+B*t+C with A=4, B=3, C=170, a=-15, b=16 and a<b. Coefficients are quadratic, linear and constant in that order. This declared polynomial is continuous on [a,b] and differentiable on (a,b); finite samples do not certify those hypotheses for an arbitrary function. Give (a, b, ordered_endpoints, closed_interval_continuity, open_interval_differentiability, certificate).

View answer and explanation(-15, 16, Yes, Yes, Yes, polynomial_continuous_on_closed_interval_and_differentiable_on_open_interval)

Audit a<b, continuity on the closed interval and differentiability on the open interval. A theorem witness must lie strictly inside (a,b); existence does not imply uniqueness. The exact result, retaining the requested order, is (-15, 16, Yes, Yes, Yes, polynomial_continuous_on_closed_interval_and_differentiable_on_open_interval).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r02-mean-value-theorem-0014/)

Mean value theorem · qf-mod41-mvt-hypothesis-audit · candidate





## 15. Continuity and Differentiability Apply on Different Interval Parts

Use the whole real polynomial f(t)=A*t^2+B*t+C with A=5, B=4, C=197, a=-16, b=17 and a<b. Coefficients are quadratic, linear and constant in that order. This declared polynomial is continuous on [a,b] and differentiable on (a,b); finite samples do not certify those hypotheses for an arbitrary function. Give (a, b, ordered_endpoints, closed_interval_continuity, open_interval_differentiability, certificate).

View answer and explanation(-16, 17, Yes, Yes, Yes, polynomial_continuous_on_closed_interval_and_differentiable_on_open_interval)

Audit a<b, continuity on the closed interval and differentiability on the open interval. A theorem witness must lie strictly inside (a,b); existence does not imply uniqueness. The exact result, retaining the requested order, is (-16, 17, Yes, Yes, Yes, polynomial_continuous_on_closed_interval_and_differentiable_on_open_interval).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r02-mean-value-theorem-0015/)

Mean value theorem · qf-mod41-mvt-hypothesis-audit · candidate





## 16. Continuity and Differentiability Apply on Different Interval Parts

Use the whole real polynomial f(t)=A*t^2+B*t+C with A=1, B=5, C=226, a=-17, b=18 and a<b. Coefficients are quadratic, linear and constant in that order. This declared polynomial is continuous on [a,b] and differentiable on (a,b); finite samples do not certify those hypotheses for an arbitrary function. Give (a, b, ordered_endpoints, closed_interval_continuity, open_interval_differentiability, certificate).

View answer and explanation(-17, 18, Yes, Yes, Yes, polynomial_continuous_on_closed_interval_and_differentiable_on_open_interval)

Audit a<b, continuity on the closed interval and differentiability on the open interval. A theorem witness must lie strictly inside (a,b); existence does not imply uniqueness. The exact result, retaining the requested order, is (-17, 18, Yes, Yes, Yes, polynomial_continuous_on_closed_interval_and_differentiable_on_open_interval).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r02-mean-value-theorem-0016/)

Mean value theorem · qf-mod41-mvt-hypothesis-audit · candidate





## 17. Continuity and Differentiability Apply on Different Interval Parts

Use the whole real polynomial f(t)=A*t^2+B*t+C with A=2, B=6, C=257, a=-18, b=19 and a<b. Coefficients are quadratic, linear and constant in that order. This declared polynomial is continuous on [a,b] and differentiable on (a,b); finite samples do not certify those hypotheses for an arbitrary function. Give (a, b, ordered_endpoints, closed_interval_continuity, open_interval_differentiability, certificate).

View answer and explanation(-18, 19, Yes, Yes, Yes, polynomial_continuous_on_closed_interval_and_differentiable_on_open_interval)

Audit a<b, continuity on the closed interval and differentiability on the open interval. A theorem witness must lie strictly inside (a,b); existence does not imply uniqueness. The exact result, retaining the requested order, is (-18, 19, Yes, Yes, Yes, polynomial_continuous_on_closed_interval_and_differentiable_on_open_interval).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r02-mean-value-theorem-0017/)

Mean value theorem · qf-mod41-mvt-hypothesis-audit · candidate





## 18. Continuity and Differentiability Apply on Different Interval Parts

Use the whole real polynomial f(t)=A*t^2+B*t+C with A=3, B=7, C=290, a=-19, b=20 and a<b. Coefficients are quadratic, linear and constant in that order. This declared polynomial is continuous on [a,b] and differentiable on (a,b); finite samples do not certify those hypotheses for an arbitrary function. Give (a, b, ordered_endpoints, closed_interval_continuity, open_interval_differentiability, certificate).

View answer and explanation(-19, 20, Yes, Yes, Yes, polynomial_continuous_on_closed_interval_and_differentiable_on_open_interval)

Audit a<b, continuity on the closed interval and differentiability on the open interval. A theorem witness must lie strictly inside (a,b); existence does not imply uniqueness. The exact result, retaining the requested order, is (-19, 20, Yes, Yes, Yes, polynomial_continuous_on_closed_interval_and_differentiable_on_open_interval).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r02-mean-value-theorem-0018/)

Mean value theorem · qf-mod41-mvt-hypothesis-audit · candidate





## 19. Continuity and Differentiability Apply on Different Interval Parts

Use the whole real polynomial f(t)=A*t^2+B*t+C with A=4, B=8, C=325, a=-20, b=21 and a<b. Coefficients are quadratic, linear and constant in that order. This declared polynomial is continuous on [a,b] and differentiable on (a,b); finite samples do not certify those hypotheses for an arbitrary function. Give (a, b, ordered_endpoints, closed_interval_continuity, open_interval_differentiability, certificate).

View answer and explanation(-20, 21, Yes, Yes, Yes, polynomial_continuous_on_closed_interval_and_differentiable_on_open_interval)

Audit a<b, continuity on the closed interval and differentiability on the open interval. A theorem witness must lie strictly inside (a,b); existence does not imply uniqueness. The exact result, retaining the requested order, is (-20, 21, Yes, Yes, Yes, polynomial_continuous_on_closed_interval_and_differentiable_on_open_interval).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r02-mean-value-theorem-0019/)

Mean value theorem · qf-mod41-mvt-hypothesis-audit · candidate





## 20. Continuity and Differentiability Apply on Different Interval Parts

Use the whole real polynomial f(t)=A*t^2+B*t+C with A=5, B=9, C=362, a=-21, b=22 and a<b. Coefficients are quadratic, linear and constant in that order. This declared polynomial is continuous on [a,b] and differentiable on (a,b); finite samples do not certify those hypotheses for an arbitrary function. Give (a, b, ordered_endpoints, closed_interval_continuity, open_interval_differentiability, certificate).

View answer and explanation(-21, 22, Yes, Yes, Yes, polynomial_continuous_on_closed_interval_and_differentiable_on_open_interval)

Audit a<b, continuity on the closed interval and differentiability on the open interval. A theorem witness must lie strictly inside (a,b); existence does not imply uniqueness. The exact result, retaining the requested order, is (-21, 22, Yes, Yes, Yes, polynomial_continuous_on_closed_interval_and_differentiable_on_open_interval).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r02-mean-value-theorem-0020/)

Mean value theorem · qf-mod41-mvt-hypothesis-audit · candidate





## 21. Continuity and Differentiability Apply on Different Interval Parts

Use the whole real polynomial f(t)=A*t^2+B*t+C with A=1, B=10, C=401, a=-22, b=23 and a<b. Coefficients are quadratic, linear and constant in that order. This declared polynomial is continuous on [a,b] and differentiable on (a,b); finite samples do not certify those hypotheses for an arbitrary function. Give (a, b, ordered_endpoints, closed_interval_continuity, open_interval_differentiability, certificate).

View answer and explanation(-22, 23, Yes, Yes, Yes, polynomial_continuous_on_closed_interval_and_differentiable_on_open_interval)

Audit a<b, continuity on the closed interval and differentiability on the open interval. A theorem witness must lie strictly inside (a,b); existence does not imply uniqueness. The exact result, retaining the requested order, is (-22, 23, Yes, Yes, Yes, polynomial_continuous_on_closed_interval_and_differentiable_on_open_interval).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r02-mean-value-theorem-0021/)

Mean value theorem · qf-mod41-mvt-hypothesis-audit · candidate





## 22. Continuity and Differentiability Apply on Different Interval Parts

Use the whole real polynomial f(t)=A*t^2+B*t+C with A=2, B=11, C=442, a=-23, b=24 and a<b. Coefficients are quadratic, linear and constant in that order. This declared polynomial is continuous on [a,b] and differentiable on (a,b); finite samples do not certify those hypotheses for an arbitrary function. Give (a, b, ordered_endpoints, closed_interval_continuity, open_interval_differentiability, certificate).

View answer and explanation(-23, 24, Yes, Yes, Yes, polynomial_continuous_on_closed_interval_and_differentiable_on_open_interval)

Audit a<b, continuity on the closed interval and differentiability on the open interval. A theorem witness must lie strictly inside (a,b); existence does not imply uniqueness. The exact result, retaining the requested order, is (-23, 24, Yes, Yes, Yes, polynomial_continuous_on_closed_interval_and_differentiable_on_open_interval).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r02-mean-value-theorem-0022/)

Mean value theorem · qf-mod41-mvt-hypothesis-audit · candidate





## 23. Continuity and Differentiability Apply on Different Interval Parts

Use the whole real polynomial f(t)=A*t^2+B*t+C with A=3, B=12, C=485, a=-24, b=25 and a<b. Coefficients are quadratic, linear and constant in that order. This declared polynomial is continuous on [a,b] and differentiable on (a,b); finite samples do not certify those hypotheses for an arbitrary function. Give (a, b, ordered_endpoints, closed_interval_continuity, open_interval_differentiability, certificate).

View answer and explanation(-24, 25, Yes, Yes, Yes, polynomial_continuous_on_closed_interval_and_differentiable_on_open_interval)

Audit a<b, continuity on the closed interval and differentiability on the open interval. A theorem witness must lie strictly inside (a,b); existence does not imply uniqueness. The exact result, retaining the requested order, is (-24, 25, Yes, Yes, Yes, polynomial_continuous_on_closed_interval_and_differentiable_on_open_interval).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r02-mean-value-theorem-0023/)

Mean value theorem · qf-mod41-mvt-hypothesis-audit · candidate





## 24. Continuity and Differentiability Apply on Different Interval Parts

Use the whole real polynomial f(t)=A*t^2+B*t+C with A=4, B=13, C=530, a=-25, b=26 and a<b. Coefficients are quadratic, linear and constant in that order. This declared polynomial is continuous on [a,b] and differentiable on (a,b); finite samples do not certify those hypotheses for an arbitrary function. Give (a, b, ordered_endpoints, closed_interval_continuity, open_interval_differentiability, certificate).

View answer and explanation(-25, 26, Yes, Yes, Yes, polynomial_continuous_on_closed_interval_and_differentiable_on_open_interval)

Audit a<b, continuity on the closed interval and differentiability on the open interval. A theorem witness must lie strictly inside (a,b); existence does not imply uniqueness. The exact result, retaining the requested order, is (-25, 26, Yes, Yes, Yes, polynomial_continuous_on_closed_interval_and_differentiable_on_open_interval).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r02-mean-value-theorem-0024/)

Mean value theorem · qf-mod41-mvt-hypothesis-audit · candidate





## 25. Continuity and Differentiability Apply on Different Interval Parts

Use the whole real polynomial f(t)=A*t^2+B*t+C with A=5, B=14, C=577, a=-26, b=27 and a<b. Coefficients are quadratic, linear and constant in that order. This declared polynomial is continuous on [a,b] and differentiable on (a,b); finite samples do not certify those hypotheses for an arbitrary function. Give (a, b, ordered_endpoints, closed_interval_continuity, open_interval_differentiability, certificate).

View answer and explanation(-26, 27, Yes, Yes, Yes, polynomial_continuous_on_closed_interval_and_differentiable_on_open_interval)

Audit a<b, continuity on the closed interval and differentiability on the open interval. A theorem witness must lie strictly inside (a,b); existence does not imply uniqueness. The exact result, retaining the requested order, is (-26, 27, Yes, Yes, Yes, polynomial_continuous_on_closed_interval_and_differentiable_on_open_interval).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r02-mean-value-theorem-0025/)

Mean value theorem · qf-mod41-mvt-hypothesis-audit · candidate





## 26. The Secant Uses Exact Endpoint Values and a Nonzero Denominator

Use the whole real polynomial f(t)=A*t^2+B*t+C with A=1, B=-10, C=1, a=-2, b=3 and a<b. Coefficients are quadratic, linear and constant in that order. This declared polynomial is continuous on [a,b] and differentiable on (a,b); finite samples do not certify those hypotheses for an arbitrary function. Give (value_at_a, value_at_b, interval_length, secant_slope, certificate).

View answer and explanation(25, -20, 5, -9, exact_endpoint_secant_slope)

Compute f(a), f(b), b−a and (f(b)−f(a))/(b−a) exactly. This is an average slope, not an asserted endpoint derivative or finite-sampling differentiability proof. The exact result, retaining the requested order, is (25, -20, 5, -9, exact_endpoint_secant_slope).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r02-mean-value-theorem-0026/)

Mean value theorem · qf-mod41-mvt-secant-slope · candidate





## 27. The Secant Uses Exact Endpoint Values and a Nonzero Denominator

Use the whole real polynomial f(t)=A*t^2+B*t+C with A=2, B=-9, C=2, a=-3, b=4 and a<b. Coefficients are quadratic, linear and constant in that order. This declared polynomial is continuous on [a,b] and differentiable on (a,b); finite samples do not certify those hypotheses for an arbitrary function. Give (value_at_a, value_at_b, interval_length, secant_slope, certificate).

View answer and explanation(47, -2, 7, -7, exact_endpoint_secant_slope)

Compute f(a), f(b), b−a and (f(b)−f(a))/(b−a) exactly. This is an average slope, not an asserted endpoint derivative or finite-sampling differentiability proof. The exact result, retaining the requested order, is (47, -2, 7, -7, exact_endpoint_secant_slope).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r02-mean-value-theorem-0027/)

Mean value theorem · qf-mod41-mvt-secant-slope · candidate





## 28. The Secant Uses Exact Endpoint Values and a Nonzero Denominator

Use the whole real polynomial f(t)=A*t^2+B*t+C with A=3, B=-8, C=5, a=-4, b=5 and a<b. Coefficients are quadratic, linear and constant in that order. This declared polynomial is continuous on [a,b] and differentiable on (a,b); finite samples do not certify those hypotheses for an arbitrary function. Give (value_at_a, value_at_b, interval_length, secant_slope, certificate).

View answer and explanation(85, 40, 9, -5, exact_endpoint_secant_slope)

Compute f(a), f(b), b−a and (f(b)−f(a))/(b−a) exactly. This is an average slope, not an asserted endpoint derivative or finite-sampling differentiability proof. The exact result, retaining the requested order, is (85, 40, 9, -5, exact_endpoint_secant_slope).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r02-mean-value-theorem-0028/)

Mean value theorem · qf-mod41-mvt-secant-slope · candidate





## 29. The Secant Uses Exact Endpoint Values and a Nonzero Denominator

Use the whole real polynomial f(t)=A*t^2+B*t+C with A=4, B=-7, C=10, a=-5, b=6 and a<b. Coefficients are quadratic, linear and constant in that order. This declared polynomial is continuous on [a,b] and differentiable on (a,b); finite samples do not certify those hypotheses for an arbitrary function. Give (value_at_a, value_at_b, interval_length, secant_slope, certificate).

View answer and explanation(145, 112, 11, -3, exact_endpoint_secant_slope)

Compute f(a), f(b), b−a and (f(b)−f(a))/(b−a) exactly. This is an average slope, not an asserted endpoint derivative or finite-sampling differentiability proof. The exact result, retaining the requested order, is (145, 112, 11, -3, exact_endpoint_secant_slope).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r02-mean-value-theorem-0029/)

Mean value theorem · qf-mod41-mvt-secant-slope · candidate





## 30. The Secant Uses Exact Endpoint Values and a Nonzero Denominator

Use the whole real polynomial f(t)=A*t^2+B*t+C with A=5, B=-6, C=17, a=-6, b=7 and a<b. Coefficients are quadratic, linear and constant in that order. This declared polynomial is continuous on [a,b] and differentiable on (a,b); finite samples do not certify those hypotheses for an arbitrary function. Give (value_at_a, value_at_b, interval_length, secant_slope, certificate).

View answer and explanation(233, 220, 13, -1, exact_endpoint_secant_slope)

Compute f(a), f(b), b−a and (f(b)−f(a))/(b−a) exactly. This is an average slope, not an asserted endpoint derivative or finite-sampling differentiability proof. The exact result, retaining the requested order, is (233, 220, 13, -1, exact_endpoint_secant_slope).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r02-mean-value-theorem-0030/)

Mean value theorem · qf-mod41-mvt-secant-slope · candidate





## 31. The Secant Uses Exact Endpoint Values and a Nonzero Denominator

Use the whole real polynomial f(t)=A*t^2+B*t+C with A=1, B=-5, C=26, a=-7, b=8 and a<b. Coefficients are quadratic, linear and constant in that order. This declared polynomial is continuous on [a,b] and differentiable on (a,b); finite samples do not certify those hypotheses for an arbitrary function. Give (value_at_a, value_at_b, interval_length, secant_slope, certificate).

View answer and explanation(110, 50, 15, -4, exact_endpoint_secant_slope)

Compute f(a), f(b), b−a and (f(b)−f(a))/(b−a) exactly. This is an average slope, not an asserted endpoint derivative or finite-sampling differentiability proof. The exact result, retaining the requested order, is (110, 50, 15, -4, exact_endpoint_secant_slope).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r02-mean-value-theorem-0031/)

Mean value theorem · qf-mod41-mvt-secant-slope · candidate





## 32. The Secant Uses Exact Endpoint Values and a Nonzero Denominator

Use the whole real polynomial f(t)=A*t^2+B*t+C with A=2, B=-4, C=37, a=-8, b=9 and a<b. Coefficients are quadratic, linear and constant in that order. This declared polynomial is continuous on [a,b] and differentiable on (a,b); finite samples do not certify those hypotheses for an arbitrary function. Give (value_at_a, value_at_b, interval_length, secant_slope, certificate).

View answer and explanation(197, 163, 17, -2, exact_endpoint_secant_slope)

Compute f(a), f(b), b−a and (f(b)−f(a))/(b−a) exactly. This is an average slope, not an asserted endpoint derivative or finite-sampling differentiability proof. The exact result, retaining the requested order, is (197, 163, 17, -2, exact_endpoint_secant_slope).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r02-mean-value-theorem-0032/)

Mean value theorem · qf-mod41-mvt-secant-slope · candidate





## 33. The Secant Uses Exact Endpoint Values and a Nonzero Denominator

Use the whole real polynomial f(t)=A*t^2+B*t+C with A=3, B=-3, C=50, a=-9, b=10 and a<b. Coefficients are quadratic, linear and constant in that order. This declared polynomial is continuous on [a,b] and differentiable on (a,b); finite samples do not certify those hypotheses for an arbitrary function. Give (value_at_a, value_at_b, interval_length, secant_slope, certificate).

View answer and explanation(320, 320, 19, 0, exact_endpoint_secant_slope)

Compute f(a), f(b), b−a and (f(b)−f(a))/(b−a) exactly. This is an average slope, not an asserted endpoint derivative or finite-sampling differentiability proof. The exact result, retaining the requested order, is (320, 320, 19, 0, exact_endpoint_secant_slope).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r02-mean-value-theorem-0033/)

Mean value theorem · qf-mod41-mvt-secant-slope · candidate





## 34. The Secant Uses Exact Endpoint Values and a Nonzero Denominator

Use the whole real polynomial f(t)=A*t^2+B*t+C with A=4, B=-2, C=65, a=-10, b=11 and a<b. Coefficients are quadratic, linear and constant in that order. This declared polynomial is continuous on [a,b] and differentiable on (a,b); finite samples do not certify those hypotheses for an arbitrary function. Give (value_at_a, value_at_b, interval_length, secant_slope, certificate).

View answer and explanation(485, 527, 21, 2, exact_endpoint_secant_slope)

Compute f(a), f(b), b−a and (f(b)−f(a))/(b−a) exactly. This is an average slope, not an asserted endpoint derivative or finite-sampling differentiability proof. The exact result, retaining the requested order, is (485, 527, 21, 2, exact_endpoint_secant_slope).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r02-mean-value-theorem-0034/)

Mean value theorem · qf-mod41-mvt-secant-slope · candidate





## 35. The Secant Uses Exact Endpoint Values and a Nonzero Denominator

Use the whole real polynomial f(t)=A*t^2+B*t+C with A=5, B=-1, C=82, a=-11, b=12 and a<b. Coefficients are quadratic, linear and constant in that order. This declared polynomial is continuous on [a,b] and differentiable on (a,b); finite samples do not certify those hypotheses for an arbitrary function. Give (value_at_a, value_at_b, interval_length, secant_slope, certificate).

View answer and explanation(698, 790, 23, 4, exact_endpoint_secant_slope)

Compute f(a), f(b), b−a and (f(b)−f(a))/(b−a) exactly. This is an average slope, not an asserted endpoint derivative or finite-sampling differentiability proof. The exact result, retaining the requested order, is (698, 790, 23, 4, exact_endpoint_secant_slope).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r02-mean-value-theorem-0035/)

Mean value theorem · qf-mod41-mvt-secant-slope · candidate





## 36. The Secant Uses Exact Endpoint Values and a Nonzero Denominator

Use the whole real polynomial f(t)=A*t^2+B*t+C with A=1, B=0, C=101, a=-12, b=13 and a<b. Coefficients are quadratic, linear and constant in that order. This declared polynomial is continuous on [a,b] and differentiable on (a,b); finite samples do not certify those hypotheses for an arbitrary function. Give (value_at_a, value_at_b, interval_length, secant_slope, certificate).

View answer and explanation(245, 270, 25, 1, exact_endpoint_secant_slope)

Compute f(a), f(b), b−a and (f(b)−f(a))/(b−a) exactly. This is an average slope, not an asserted endpoint derivative or finite-sampling differentiability proof. The exact result, retaining the requested order, is (245, 270, 25, 1, exact_endpoint_secant_slope).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r02-mean-value-theorem-0036/)

Mean value theorem · qf-mod41-mvt-secant-slope · candidate





## 37. The Secant Uses Exact Endpoint Values and a Nonzero Denominator

Use the whole real polynomial f(t)=A*t^2+B*t+C with A=2, B=1, C=122, a=-13, b=14 and a<b. Coefficients are quadratic, linear and constant in that order. This declared polynomial is continuous on [a,b] and differentiable on (a,b); finite samples do not certify those hypotheses for an arbitrary function. Give (value_at_a, value_at_b, interval_length, secant_slope, certificate).

View answer and explanation(447, 528, 27, 3, exact_endpoint_secant_slope)

Compute f(a), f(b), b−a and (f(b)−f(a))/(b−a) exactly. This is an average slope, not an asserted endpoint derivative or finite-sampling differentiability proof. The exact result, retaining the requested order, is (447, 528, 27, 3, exact_endpoint_secant_slope).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r02-mean-value-theorem-0037/)

Mean value theorem · qf-mod41-mvt-secant-slope · candidate





## 38. The Secant Uses Exact Endpoint Values and a Nonzero Denominator

Use the whole real polynomial f(t)=A*t^2+B*t+C with A=3, B=2, C=145, a=-14, b=15 and a<b. Coefficients are quadratic, linear and constant in that order. This declared polynomial is continuous on [a,b] and differentiable on (a,b); finite samples do not certify those hypotheses for an arbitrary function. Give (value_at_a, value_at_b, interval_length, secant_slope, certificate).

View answer and explanation(705, 850, 29, 5, exact_endpoint_secant_slope)

Compute f(a), f(b), b−a and (f(b)−f(a))/(b−a) exactly. This is an average slope, not an asserted endpoint derivative or finite-sampling differentiability proof. The exact result, retaining the requested order, is (705, 850, 29, 5, exact_endpoint_secant_slope).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r02-mean-value-theorem-0038/)

Mean value theorem · qf-mod41-mvt-secant-slope · candidate





## 39. The Secant Uses Exact Endpoint Values and a Nonzero Denominator

Use the whole real polynomial f(t)=A*t^2+B*t+C with A=4, B=3, C=170, a=-15, b=16 and a<b. Coefficients are quadratic, linear and constant in that order. This declared polynomial is continuous on [a,b] and differentiable on (a,b); finite samples do not certify those hypotheses for an arbitrary function. Give (value_at_a, value_at_b, interval_length, secant_slope, certificate).

View answer and explanation(1025, 1242, 31, 7, exact_endpoint_secant_slope)

Compute f(a), f(b), b−a and (f(b)−f(a))/(b−a) exactly. This is an average slope, not an asserted endpoint derivative or finite-sampling differentiability proof. The exact result, retaining the requested order, is (1025, 1242, 31, 7, exact_endpoint_secant_slope).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r02-mean-value-theorem-0039/)

Mean value theorem · qf-mod41-mvt-secant-slope · candidate





## 40. The Secant Uses Exact Endpoint Values and a Nonzero Denominator

Use the whole real polynomial f(t)=A*t^2+B*t+C with A=5, B=4, C=197, a=-16, b=17 and a<b. Coefficients are quadratic, linear and constant in that order. This declared polynomial is continuous on [a,b] and differentiable on (a,b); finite samples do not certify those hypotheses for an arbitrary function. Give (value_at_a, value_at_b, interval_length, secant_slope, certificate).

View answer and explanation(1413, 1710, 33, 9, exact_endpoint_secant_slope)

Compute f(a), f(b), b−a and (f(b)−f(a))/(b−a) exactly. This is an average slope, not an asserted endpoint derivative or finite-sampling differentiability proof. The exact result, retaining the requested order, is (1413, 1710, 33, 9, exact_endpoint_secant_slope).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r02-mean-value-theorem-0040/)

Mean value theorem · qf-mod41-mvt-secant-slope · candidate





## 41. The Secant Uses Exact Endpoint Values and a Nonzero Denominator

Use the whole real polynomial f(t)=A*t^2+B*t+C with A=1, B=5, C=226, a=-17, b=18 and a<b. Coefficients are quadratic, linear and constant in that order. This declared polynomial is continuous on [a,b] and differentiable on (a,b); finite samples do not certify those hypotheses for an arbitrary function. Give (value_at_a, value_at_b, interval_length, secant_slope, certificate).

View answer and explanation(430, 640, 35, 6, exact_endpoint_secant_slope)

Compute f(a), f(b), b−a and (f(b)−f(a))/(b−a) exactly. This is an average slope, not an asserted endpoint derivative or finite-sampling differentiability proof. The exact result, retaining the requested order, is (430, 640, 35, 6, exact_endpoint_secant_slope).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r02-mean-value-theorem-0041/)

Mean value theorem · qf-mod41-mvt-secant-slope · candidate





## 42. The Secant Uses Exact Endpoint Values and a Nonzero Denominator

Use the whole real polynomial f(t)=A*t^2+B*t+C with A=2, B=6, C=257, a=-18, b=19 and a<b. Coefficients are quadratic, linear and constant in that order. This declared polynomial is continuous on [a,b] and differentiable on (a,b); finite samples do not certify those hypotheses for an arbitrary function. Give (value_at_a, value_at_b, interval_length, secant_slope, certificate).

View answer and explanation(797, 1093, 37, 8, exact_endpoint_secant_slope)

Compute f(a), f(b), b−a and (f(b)−f(a))/(b−a) exactly. This is an average slope, not an asserted endpoint derivative or finite-sampling differentiability proof. The exact result, retaining the requested order, is (797, 1093, 37, 8, exact_endpoint_secant_slope).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r02-mean-value-theorem-0042/)

Mean value theorem · qf-mod41-mvt-secant-slope · candidate





## 43. The Secant Uses Exact Endpoint Values and a Nonzero Denominator

Use the whole real polynomial f(t)=A*t^2+B*t+C with A=3, B=7, C=290, a=-19, b=20 and a<b. Coefficients are quadratic, linear and constant in that order. This declared polynomial is continuous on [a,b] and differentiable on (a,b); finite samples do not certify those hypotheses for an arbitrary function. Give (value_at_a, value_at_b, interval_length, secant_slope, certificate).

View answer and explanation(1240, 1630, 39, 10, exact_endpoint_secant_slope)

Compute f(a), f(b), b−a and (f(b)−f(a))/(b−a) exactly. This is an average slope, not an asserted endpoint derivative or finite-sampling differentiability proof. The exact result, retaining the requested order, is (1240, 1630, 39, 10, exact_endpoint_secant_slope).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r02-mean-value-theorem-0043/)

Mean value theorem · qf-mod41-mvt-secant-slope · candidate





## 44. The Secant Uses Exact Endpoint Values and a Nonzero Denominator

Use the whole real polynomial f(t)=A*t^2+B*t+C with A=4, B=8, C=325, a=-20, b=21 and a<b. Coefficients are quadratic, linear and constant in that order. This declared polynomial is continuous on [a,b] and differentiable on (a,b); finite samples do not certify those hypotheses for an arbitrary function. Give (value_at_a, value_at_b, interval_length, secant_slope, certificate).

View answer and explanation(1765, 2257, 41, 12, exact_endpoint_secant_slope)

Compute f(a), f(b), b−a and (f(b)−f(a))/(b−a) exactly. This is an average slope, not an asserted endpoint derivative or finite-sampling differentiability proof. The exact result, retaining the requested order, is (1765, 2257, 41, 12, exact_endpoint_secant_slope).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r02-mean-value-theorem-0044/)

Mean value theorem · qf-mod41-mvt-secant-slope · candidate





## 45. The Secant Uses Exact Endpoint Values and a Nonzero Denominator

Use the whole real polynomial f(t)=A*t^2+B*t+C with A=5, B=9, C=362, a=-21, b=22 and a<b. Coefficients are quadratic, linear and constant in that order. This declared polynomial is continuous on [a,b] and differentiable on (a,b); finite samples do not certify those hypotheses for an arbitrary function. Give (value_at_a, value_at_b, interval_length, secant_slope, certificate).

View answer and explanation(2378, 2980, 43, 14, exact_endpoint_secant_slope)

Compute f(a), f(b), b−a and (f(b)−f(a))/(b−a) exactly. This is an average slope, not an asserted endpoint derivative or finite-sampling differentiability proof. The exact result, retaining the requested order, is (2378, 2980, 43, 14, exact_endpoint_secant_slope).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r02-mean-value-theorem-0045/)

Mean value theorem · qf-mod41-mvt-secant-slope · candidate





## 46. The Secant Uses Exact Endpoint Values and a Nonzero Denominator

Use the whole real polynomial f(t)=A*t^2+B*t+C with A=1, B=10, C=401, a=-22, b=23 and a<b. Coefficients are quadratic, linear and constant in that order. This declared polynomial is continuous on [a,b] and differentiable on (a,b); finite samples do not certify those hypotheses for an arbitrary function. Give (value_at_a, value_at_b, interval_length, secant_slope, certificate).

View answer and explanation(665, 1160, 45, 11, exact_endpoint_secant_slope)

Compute f(a), f(b), b−a and (f(b)−f(a))/(b−a) exactly. This is an average slope, not an asserted endpoint derivative or finite-sampling differentiability proof. The exact result, retaining the requested order, is (665, 1160, 45, 11, exact_endpoint_secant_slope).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r02-mean-value-theorem-0046/)

Mean value theorem · qf-mod41-mvt-secant-slope · candidate





## 47. The Secant Uses Exact Endpoint Values and a Nonzero Denominator

Use the whole real polynomial f(t)=A*t^2+B*t+C with A=2, B=11, C=442, a=-23, b=24 and a<b. Coefficients are quadratic, linear and constant in that order. This declared polynomial is continuous on [a,b] and differentiable on (a,b); finite samples do not certify those hypotheses for an arbitrary function. Give (value_at_a, value_at_b, interval_length, secant_slope, certificate).

View answer and explanation(1247, 1858, 47, 13, exact_endpoint_secant_slope)

Compute f(a), f(b), b−a and (f(b)−f(a))/(b−a) exactly. This is an average slope, not an asserted endpoint derivative or finite-sampling differentiability proof. The exact result, retaining the requested order, is (1247, 1858, 47, 13, exact_endpoint_secant_slope).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r02-mean-value-theorem-0047/)

Mean value theorem · qf-mod41-mvt-secant-slope · candidate





## 48. The Secant Uses Exact Endpoint Values and a Nonzero Denominator

Use the whole real polynomial f(t)=A*t^2+B*t+C with A=3, B=12, C=485, a=-24, b=25 and a<b. Coefficients are quadratic, linear and constant in that order. This declared polynomial is continuous on [a,b] and differentiable on (a,b); finite samples do not certify those hypotheses for an arbitrary function. Give (value_at_a, value_at_b, interval_length, secant_slope, certificate).

View answer and explanation(1925, 2660, 49, 15, exact_endpoint_secant_slope)

Compute f(a), f(b), b−a and (f(b)−f(a))/(b−a) exactly. This is an average slope, not an asserted endpoint derivative or finite-sampling differentiability proof. The exact result, retaining the requested order, is (1925, 2660, 49, 15, exact_endpoint_secant_slope).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r02-mean-value-theorem-0048/)

Mean value theorem · qf-mod41-mvt-secant-slope · candidate





## 49. The Secant Uses Exact Endpoint Values and a Nonzero Denominator

Use the whole real polynomial f(t)=A*t^2+B*t+C with A=4, B=13, C=530, a=-25, b=26 and a<b. Coefficients are quadratic, linear and constant in that order. This declared polynomial is continuous on [a,b] and differentiable on (a,b); finite samples do not certify those hypotheses for an arbitrary function. Give (value_at_a, value_at_b, interval_length, secant_slope, certificate).

View answer and explanation(2705, 3572, 51, 17, exact_endpoint_secant_slope)

Compute f(a), f(b), b−a and (f(b)−f(a))/(b−a) exactly. This is an average slope, not an asserted endpoint derivative or finite-sampling differentiability proof. The exact result, retaining the requested order, is (2705, 3572, 51, 17, exact_endpoint_secant_slope).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r02-mean-value-theorem-0049/)

Mean value theorem · qf-mod41-mvt-secant-slope · candidate





## 50. The Secant Uses Exact Endpoint Values and a Nonzero Denominator

Use the whole real polynomial f(t)=A*t^2+B*t+C with A=5, B=14, C=577, a=-26, b=27 and a<b. Coefficients are quadratic, linear and constant in that order. This declared polynomial is continuous on [a,b] and differentiable on (a,b); finite samples do not certify those hypotheses for an arbitrary function. Give (value_at_a, value_at_b, interval_length, secant_slope, certificate).

View answer and explanation(3593, 4600, 53, 19, exact_endpoint_secant_slope)

Compute f(a), f(b), b−a and (f(b)−f(a))/(b−a) exactly. This is an average slope, not an asserted endpoint derivative or finite-sampling differentiability proof. The exact result, retaining the requested order, is (3593, 4600, 53, 19, exact_endpoint_secant_slope).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r02-mean-value-theorem-0050/)

Mean value theorem · qf-mod41-mvt-secant-slope · candidate





## 51. The Quadratic Midpoint Is a Strictly Interior MVT Witness

Use the whole real polynomial f(t)=A*t^2+B*t+C with A=1, B=-10, C=1, a=-2, b=3 and a<b. Coefficients are quadratic, linear and constant in that order. This declared polynomial is continuous on [a,b] and differentiable on (a,b); finite samples do not certify those hypotheses for an arbitrary function. Give (c, strictly_interior, derivative_at_c, secant_slope, equal, certificate).

View answer and explanation(1/2, Yes, -9, -9, Yes, quadratic_midpoint_witness)

Take c=(a+b)/2, verify a<c<b, and compare 2*A*c+B with the exact secant slope. For A=0 this affine/constant case has many witnesses; the midpoint is not a uniqueness claim. The exact result, retaining the requested order, is (1/2, Yes, -9, -9, Yes, quadratic_midpoint_witness).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r02-mean-value-theorem-0051/)

Mean value theorem · qf-mod41-mvt-polynomial-witness · candidate





## 52. The Quadratic Midpoint Is a Strictly Interior MVT Witness

Use the whole real polynomial f(t)=A*t^2+B*t+C with A=2, B=-9, C=2, a=-3, b=4 and a<b. Coefficients are quadratic, linear and constant in that order. This declared polynomial is continuous on [a,b] and differentiable on (a,b); finite samples do not certify those hypotheses for an arbitrary function. Give (c, strictly_interior, derivative_at_c, secant_slope, equal, certificate).

View answer and explanation(1/2, Yes, -7, -7, Yes, quadratic_midpoint_witness)

Take c=(a+b)/2, verify a<c<b, and compare 2*A*c+B with the exact secant slope. For A=0 this affine/constant case has many witnesses; the midpoint is not a uniqueness claim. The exact result, retaining the requested order, is (1/2, Yes, -7, -7, Yes, quadratic_midpoint_witness).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r02-mean-value-theorem-0052/)

Mean value theorem · qf-mod41-mvt-polynomial-witness · candidate





## 53. The Quadratic Midpoint Is a Strictly Interior MVT Witness

Use the whole real polynomial f(t)=A*t^2+B*t+C with A=3, B=-8, C=5, a=-4, b=5 and a<b. Coefficients are quadratic, linear and constant in that order. This declared polynomial is continuous on [a,b] and differentiable on (a,b); finite samples do not certify those hypotheses for an arbitrary function. Give (c, strictly_interior, derivative_at_c, secant_slope, equal, certificate).

View answer and explanation(1/2, Yes, -5, -5, Yes, quadratic_midpoint_witness)

Take c=(a+b)/2, verify a<c<b, and compare 2*A*c+B with the exact secant slope. For A=0 this affine/constant case has many witnesses; the midpoint is not a uniqueness claim. The exact result, retaining the requested order, is (1/2, Yes, -5, -5, Yes, quadratic_midpoint_witness).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r02-mean-value-theorem-0053/)

Mean value theorem · qf-mod41-mvt-polynomial-witness · candidate





## 54. The Quadratic Midpoint Is a Strictly Interior MVT Witness

Use the whole real polynomial f(t)=A*t^2+B*t+C with A=4, B=-7, C=10, a=-5, b=6 and a<b. Coefficients are quadratic, linear and constant in that order. This declared polynomial is continuous on [a,b] and differentiable on (a,b); finite samples do not certify those hypotheses for an arbitrary function. Give (c, strictly_interior, derivative_at_c, secant_slope, equal, certificate).

View answer and explanation(1/2, Yes, -3, -3, Yes, quadratic_midpoint_witness)

Take c=(a+b)/2, verify a<c<b, and compare 2*A*c+B with the exact secant slope. For A=0 this affine/constant case has many witnesses; the midpoint is not a uniqueness claim. The exact result, retaining the requested order, is (1/2, Yes, -3, -3, Yes, quadratic_midpoint_witness).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r02-mean-value-theorem-0054/)

Mean value theorem · qf-mod41-mvt-polynomial-witness · candidate





## 55. The Quadratic Midpoint Is a Strictly Interior MVT Witness

Use the whole real polynomial f(t)=A*t^2+B*t+C with A=5, B=-6, C=17, a=-6, b=7 and a<b. Coefficients are quadratic, linear and constant in that order. This declared polynomial is continuous on [a,b] and differentiable on (a,b); finite samples do not certify those hypotheses for an arbitrary function. Give (c, strictly_interior, derivative_at_c, secant_slope, equal, certificate).

View answer and explanation(1/2, Yes, -1, -1, Yes, quadratic_midpoint_witness)

Take c=(a+b)/2, verify a<c<b, and compare 2*A*c+B with the exact secant slope. For A=0 this affine/constant case has many witnesses; the midpoint is not a uniqueness claim. The exact result, retaining the requested order, is (1/2, Yes, -1, -1, Yes, quadratic_midpoint_witness).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r02-mean-value-theorem-0055/)

Mean value theorem · qf-mod41-mvt-polynomial-witness · candidate





## 56. The Quadratic Midpoint Is a Strictly Interior MVT Witness

Use the whole real polynomial f(t)=A*t^2+B*t+C with A=1, B=-5, C=26, a=-7, b=8 and a<b. Coefficients are quadratic, linear and constant in that order. This declared polynomial is continuous on [a,b] and differentiable on (a,b); finite samples do not certify those hypotheses for an arbitrary function. Give (c, strictly_interior, derivative_at_c, secant_slope, equal, certificate).

View answer and explanation(1/2, Yes, -4, -4, Yes, quadratic_midpoint_witness)

Take c=(a+b)/2, verify a<c<b, and compare 2*A*c+B with the exact secant slope. For A=0 this affine/constant case has many witnesses; the midpoint is not a uniqueness claim. The exact result, retaining the requested order, is (1/2, Yes, -4, -4, Yes, quadratic_midpoint_witness).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r02-mean-value-theorem-0056/)

Mean value theorem · qf-mod41-mvt-polynomial-witness · candidate





## 57. The Quadratic Midpoint Is a Strictly Interior MVT Witness

Use the whole real polynomial f(t)=A*t^2+B*t+C with A=2, B=-4, C=37, a=-8, b=9 and a<b. Coefficients are quadratic, linear and constant in that order. This declared polynomial is continuous on [a,b] and differentiable on (a,b); finite samples do not certify those hypotheses for an arbitrary function. Give (c, strictly_interior, derivative_at_c, secant_slope, equal, certificate).

View answer and explanation(1/2, Yes, -2, -2, Yes, quadratic_midpoint_witness)

Take c=(a+b)/2, verify a<c<b, and compare 2*A*c+B with the exact secant slope. For A=0 this affine/constant case has many witnesses; the midpoint is not a uniqueness claim. The exact result, retaining the requested order, is (1/2, Yes, -2, -2, Yes, quadratic_midpoint_witness).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r02-mean-value-theorem-0057/)

Mean value theorem · qf-mod41-mvt-polynomial-witness · candidate





## 58. The Quadratic Midpoint Is a Strictly Interior MVT Witness

Use the whole real polynomial f(t)=A*t^2+B*t+C with A=3, B=-3, C=50, a=-9, b=10 and a<b. Coefficients are quadratic, linear and constant in that order. This declared polynomial is continuous on [a,b] and differentiable on (a,b); finite samples do not certify those hypotheses for an arbitrary function. Give (c, strictly_interior, derivative_at_c, secant_slope, equal, certificate).

View answer and explanation(1/2, Yes, 0, 0, Yes, quadratic_midpoint_witness)

Take c=(a+b)/2, verify a<c<b, and compare 2*A*c+B with the exact secant slope. For A=0 this affine/constant case has many witnesses; the midpoint is not a uniqueness claim. The exact result, retaining the requested order, is (1/2, Yes, 0, 0, Yes, quadratic_midpoint_witness).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r02-mean-value-theorem-0058/)

Mean value theorem · qf-mod41-mvt-polynomial-witness · candidate





## 59. The Quadratic Midpoint Is a Strictly Interior MVT Witness

Use the whole real polynomial f(t)=A*t^2+B*t+C with A=4, B=-2, C=65, a=-10, b=11 and a<b. Coefficients are quadratic, linear and constant in that order. This declared polynomial is continuous on [a,b] and differentiable on (a,b); finite samples do not certify those hypotheses for an arbitrary function. Give (c, strictly_interior, derivative_at_c, secant_slope, equal, certificate).

View answer and explanation(1/2, Yes, 2, 2, Yes, quadratic_midpoint_witness)

Take c=(a+b)/2, verify a<c<b, and compare 2*A*c+B with the exact secant slope. For A=0 this affine/constant case has many witnesses; the midpoint is not a uniqueness claim. The exact result, retaining the requested order, is (1/2, Yes, 2, 2, Yes, quadratic_midpoint_witness).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r02-mean-value-theorem-0059/)

Mean value theorem · qf-mod41-mvt-polynomial-witness · candidate





## 60. The Quadratic Midpoint Is a Strictly Interior MVT Witness

Use the whole real polynomial f(t)=A*t^2+B*t+C with A=5, B=-1, C=82, a=-11, b=12 and a<b. Coefficients are quadratic, linear and constant in that order. This declared polynomial is continuous on [a,b] and differentiable on (a,b); finite samples do not certify those hypotheses for an arbitrary function. Give (c, strictly_interior, derivative_at_c, secant_slope, equal, certificate).

View answer and explanation(1/2, Yes, 4, 4, Yes, quadratic_midpoint_witness)

Take c=(a+b)/2, verify a<c<b, and compare 2*A*c+B with the exact secant slope. For A=0 this affine/constant case has many witnesses; the midpoint is not a uniqueness claim. The exact result, retaining the requested order, is (1/2, Yes, 4, 4, Yes, quadratic_midpoint_witness).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r02-mean-value-theorem-0060/)

Mean value theorem · qf-mod41-mvt-polynomial-witness · candidate





## 61. The Quadratic Midpoint Is a Strictly Interior MVT Witness

Use the whole real polynomial f(t)=A*t^2+B*t+C with A=1, B=0, C=101, a=-12, b=13 and a<b. Coefficients are quadratic, linear and constant in that order. This declared polynomial is continuous on [a,b] and differentiable on (a,b); finite samples do not certify those hypotheses for an arbitrary function. Give (c, strictly_interior, derivative_at_c, secant_slope, equal, certificate).

View answer and explanation(1/2, Yes, 1, 1, Yes, quadratic_midpoint_witness)

Take c=(a+b)/2, verify a<c<b, and compare 2*A*c+B with the exact secant slope. For A=0 this affine/constant case has many witnesses; the midpoint is not a uniqueness claim. The exact result, retaining the requested order, is (1/2, Yes, 1, 1, Yes, quadratic_midpoint_witness).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r02-mean-value-theorem-0061/)

Mean value theorem · qf-mod41-mvt-polynomial-witness · candidate





## 62. The Quadratic Midpoint Is a Strictly Interior MVT Witness

Use the whole real polynomial f(t)=A*t^2+B*t+C with A=2, B=1, C=122, a=-13, b=14 and a<b. Coefficients are quadratic, linear and constant in that order. This declared polynomial is continuous on [a,b] and differentiable on (a,b); finite samples do not certify those hypotheses for an arbitrary function. Give (c, strictly_interior, derivative_at_c, secant_slope, equal, certificate).

View answer and explanation(1/2, Yes, 3, 3, Yes, quadratic_midpoint_witness)

Take c=(a+b)/2, verify a<c<b, and compare 2*A*c+B with the exact secant slope. For A=0 this affine/constant case has many witnesses; the midpoint is not a uniqueness claim. The exact result, retaining the requested order, is (1/2, Yes, 3, 3, Yes, quadratic_midpoint_witness).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r02-mean-value-theorem-0062/)

Mean value theorem · qf-mod41-mvt-polynomial-witness · candidate





## 63. The Quadratic Midpoint Is a Strictly Interior MVT Witness

Use the whole real polynomial f(t)=A*t^2+B*t+C with A=3, B=2, C=145, a=-14, b=15 and a<b. Coefficients are quadratic, linear and constant in that order. This declared polynomial is continuous on [a,b] and differentiable on (a,b); finite samples do not certify those hypotheses for an arbitrary function. Give (c, strictly_interior, derivative_at_c, secant_slope, equal, certificate).

View answer and explanation(1/2, Yes, 5, 5, Yes, quadratic_midpoint_witness)

Take c=(a+b)/2, verify a<c<b, and compare 2*A*c+B with the exact secant slope. For A=0 this affine/constant case has many witnesses; the midpoint is not a uniqueness claim. The exact result, retaining the requested order, is (1/2, Yes, 5, 5, Yes, quadratic_midpoint_witness).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r02-mean-value-theorem-0063/)

Mean value theorem · qf-mod41-mvt-polynomial-witness · candidate





## 64. The Quadratic Midpoint Is a Strictly Interior MVT Witness

Use the whole real polynomial f(t)=A*t^2+B*t+C with A=4, B=3, C=170, a=-15, b=16 and a<b. Coefficients are quadratic, linear and constant in that order. This declared polynomial is continuous on [a,b] and differentiable on (a,b); finite samples do not certify those hypotheses for an arbitrary function. Give (c, strictly_interior, derivative_at_c, secant_slope, equal, certificate).

View answer and explanation(1/2, Yes, 7, 7, Yes, quadratic_midpoint_witness)

Take c=(a+b)/2, verify a<c<b, and compare 2*A*c+B with the exact secant slope. For A=0 this affine/constant case has many witnesses; the midpoint is not a uniqueness claim. The exact result, retaining the requested order, is (1/2, Yes, 7, 7, Yes, quadratic_midpoint_witness).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r02-mean-value-theorem-0064/)

Mean value theorem · qf-mod41-mvt-polynomial-witness · candidate





## 65. The Quadratic Midpoint Is a Strictly Interior MVT Witness

Use the whole real polynomial f(t)=A*t^2+B*t+C with A=5, B=4, C=197, a=-16, b=17 and a<b. Coefficients are quadratic, linear and constant in that order. This declared polynomial is continuous on [a,b] and differentiable on (a,b); finite samples do not certify those hypotheses for an arbitrary function. Give (c, strictly_interior, derivative_at_c, secant_slope, equal, certificate).

View answer and explanation(1/2, Yes, 9, 9, Yes, quadratic_midpoint_witness)

Take c=(a+b)/2, verify a<c<b, and compare 2*A*c+B with the exact secant slope. For A=0 this affine/constant case has many witnesses; the midpoint is not a uniqueness claim. The exact result, retaining the requested order, is (1/2, Yes, 9, 9, Yes, quadratic_midpoint_witness).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r02-mean-value-theorem-0065/)

Mean value theorem · qf-mod41-mvt-polynomial-witness · candidate





## 66. The Quadratic Midpoint Is a Strictly Interior MVT Witness

Use the whole real polynomial f(t)=A*t^2+B*t+C with A=1, B=5, C=226, a=-17, b=18 and a<b. Coefficients are quadratic, linear and constant in that order. This declared polynomial is continuous on [a,b] and differentiable on (a,b); finite samples do not certify those hypotheses for an arbitrary function. Give (c, strictly_interior, derivative_at_c, secant_slope, equal, certificate).

View answer and explanation(1/2, Yes, 6, 6, Yes, quadratic_midpoint_witness)

Take c=(a+b)/2, verify a<c<b, and compare 2*A*c+B with the exact secant slope. For A=0 this affine/constant case has many witnesses; the midpoint is not a uniqueness claim. The exact result, retaining the requested order, is (1/2, Yes, 6, 6, Yes, quadratic_midpoint_witness).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r02-mean-value-theorem-0066/)

Mean value theorem · qf-mod41-mvt-polynomial-witness · candidate





## 67. The Quadratic Midpoint Is a Strictly Interior MVT Witness

Use the whole real polynomial f(t)=A*t^2+B*t+C with A=2, B=6, C=257, a=-18, b=19 and a<b. Coefficients are quadratic, linear and constant in that order. This declared polynomial is continuous on [a,b] and differentiable on (a,b); finite samples do not certify those hypotheses for an arbitrary function. Give (c, strictly_interior, derivative_at_c, secant_slope, equal, certificate).

View answer and explanation(1/2, Yes, 8, 8, Yes, quadratic_midpoint_witness)

Take c=(a+b)/2, verify a<c<b, and compare 2*A*c+B with the exact secant slope. For A=0 this affine/constant case has many witnesses; the midpoint is not a uniqueness claim. The exact result, retaining the requested order, is (1/2, Yes, 8, 8, Yes, quadratic_midpoint_witness).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r02-mean-value-theorem-0067/)

Mean value theorem · qf-mod41-mvt-polynomial-witness · candidate





## 68. The Quadratic Midpoint Is a Strictly Interior MVT Witness

Use the whole real polynomial f(t)=A*t^2+B*t+C with A=3, B=7, C=290, a=-19, b=20 and a<b. Coefficients are quadratic, linear and constant in that order. This declared polynomial is continuous on [a,b] and differentiable on (a,b); finite samples do not certify those hypotheses for an arbitrary function. Give (c, strictly_interior, derivative_at_c, secant_slope, equal, certificate).

View answer and explanation(1/2, Yes, 10, 10, Yes, quadratic_midpoint_witness)

Take c=(a+b)/2, verify a<c<b, and compare 2*A*c+B with the exact secant slope. For A=0 this affine/constant case has many witnesses; the midpoint is not a uniqueness claim. The exact result, retaining the requested order, is (1/2, Yes, 10, 10, Yes, quadratic_midpoint_witness).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r02-mean-value-theorem-0068/)

Mean value theorem · qf-mod41-mvt-polynomial-witness · candidate





## 69. The Quadratic Midpoint Is a Strictly Interior MVT Witness

Use the whole real polynomial f(t)=A*t^2+B*t+C with A=4, B=8, C=325, a=-20, b=21 and a<b. Coefficients are quadratic, linear and constant in that order. This declared polynomial is continuous on [a,b] and differentiable on (a,b); finite samples do not certify those hypotheses for an arbitrary function. Give (c, strictly_interior, derivative_at_c, secant_slope, equal, certificate).

View answer and explanation(1/2, Yes, 12, 12, Yes, quadratic_midpoint_witness)

Take c=(a+b)/2, verify a<c<b, and compare 2*A*c+B with the exact secant slope. For A=0 this affine/constant case has many witnesses; the midpoint is not a uniqueness claim. The exact result, retaining the requested order, is (1/2, Yes, 12, 12, Yes, quadratic_midpoint_witness).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r02-mean-value-theorem-0069/)

Mean value theorem · qf-mod41-mvt-polynomial-witness · candidate





## 70. The Quadratic Midpoint Is a Strictly Interior MVT Witness

Use the whole real polynomial f(t)=A*t^2+B*t+C with A=5, B=9, C=362, a=-21, b=22 and a<b. Coefficients are quadratic, linear and constant in that order. This declared polynomial is continuous on [a,b] and differentiable on (a,b); finite samples do not certify those hypotheses for an arbitrary function. Give (c, strictly_interior, derivative_at_c, secant_slope, equal, certificate).

View answer and explanation(1/2, Yes, 14, 14, Yes, quadratic_midpoint_witness)

Take c=(a+b)/2, verify a<c<b, and compare 2*A*c+B with the exact secant slope. For A=0 this affine/constant case has many witnesses; the midpoint is not a uniqueness claim. The exact result, retaining the requested order, is (1/2, Yes, 14, 14, Yes, quadratic_midpoint_witness).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r02-mean-value-theorem-0070/)

Mean value theorem · qf-mod41-mvt-polynomial-witness · candidate





## 71. The Quadratic Midpoint Is a Strictly Interior MVT Witness

Use the whole real polynomial f(t)=A*t^2+B*t+C with A=1, B=10, C=401, a=-22, b=23 and a<b. Coefficients are quadratic, linear and constant in that order. This declared polynomial is continuous on [a,b] and differentiable on (a,b); finite samples do not certify those hypotheses for an arbitrary function. Give (c, strictly_interior, derivative_at_c, secant_slope, equal, certificate).

View answer and explanation(1/2, Yes, 11, 11, Yes, quadratic_midpoint_witness)

Take c=(a+b)/2, verify a<c<b, and compare 2*A*c+B with the exact secant slope. For A=0 this affine/constant case has many witnesses; the midpoint is not a uniqueness claim. The exact result, retaining the requested order, is (1/2, Yes, 11, 11, Yes, quadratic_midpoint_witness).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r02-mean-value-theorem-0071/)

Mean value theorem · qf-mod41-mvt-polynomial-witness · candidate





## 72. The Quadratic Midpoint Is a Strictly Interior MVT Witness

Use the whole real polynomial f(t)=A*t^2+B*t+C with A=2, B=11, C=442, a=-23, b=24 and a<b. Coefficients are quadratic, linear and constant in that order. This declared polynomial is continuous on [a,b] and differentiable on (a,b); finite samples do not certify those hypotheses for an arbitrary function. Give (c, strictly_interior, derivative_at_c, secant_slope, equal, certificate).

View answer and explanation(1/2, Yes, 13, 13, Yes, quadratic_midpoint_witness)

Take c=(a+b)/2, verify a<c<b, and compare 2*A*c+B with the exact secant slope. For A=0 this affine/constant case has many witnesses; the midpoint is not a uniqueness claim. The exact result, retaining the requested order, is (1/2, Yes, 13, 13, Yes, quadratic_midpoint_witness).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r02-mean-value-theorem-0072/)

Mean value theorem · qf-mod41-mvt-polynomial-witness · candidate





## 73. The Quadratic Midpoint Is a Strictly Interior MVT Witness

Use the whole real polynomial f(t)=A*t^2+B*t+C with A=3, B=12, C=485, a=-24, b=25 and a<b. Coefficients are quadratic, linear and constant in that order. This declared polynomial is continuous on [a,b] and differentiable on (a,b); finite samples do not certify those hypotheses for an arbitrary function. Give (c, strictly_interior, derivative_at_c, secant_slope, equal, certificate).

View answer and explanation(1/2, Yes, 15, 15, Yes, quadratic_midpoint_witness)

Take c=(a+b)/2, verify a<c<b, and compare 2*A*c+B with the exact secant slope. For A=0 this affine/constant case has many witnesses; the midpoint is not a uniqueness claim. The exact result, retaining the requested order, is (1/2, Yes, 15, 15, Yes, quadratic_midpoint_witness).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r02-mean-value-theorem-0073/)

Mean value theorem · qf-mod41-mvt-polynomial-witness · candidate





## 74. The Quadratic Midpoint Is a Strictly Interior MVT Witness

Use the whole real polynomial f(t)=A*t^2+B*t+C with A=4, B=13, C=530, a=-25, b=26 and a<b. Coefficients are quadratic, linear and constant in that order. This declared polynomial is continuous on [a,b] and differentiable on (a,b); finite samples do not certify those hypotheses for an arbitrary function. Give (c, strictly_interior, derivative_at_c, secant_slope, equal, certificate).

View answer and explanation(1/2, Yes, 17, 17, Yes, quadratic_midpoint_witness)

Take c=(a+b)/2, verify a<c<b, and compare 2*A*c+B with the exact secant slope. For A=0 this affine/constant case has many witnesses; the midpoint is not a uniqueness claim. The exact result, retaining the requested order, is (1/2, Yes, 17, 17, Yes, quadratic_midpoint_witness).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r02-mean-value-theorem-0074/)

Mean value theorem · qf-mod41-mvt-polynomial-witness · candidate





## 75. The Quadratic Midpoint Is a Strictly Interior MVT Witness

Use the whole real polynomial f(t)=A*t^2+B*t+C with A=5, B=14, C=577, a=-26, b=27 and a<b. Coefficients are quadratic, linear and constant in that order. This declared polynomial is continuous on [a,b] and differentiable on (a,b); finite samples do not certify those hypotheses for an arbitrary function. Give (c, strictly_interior, derivative_at_c, secant_slope, equal, certificate).

View answer and explanation(1/2, Yes, 19, 19, Yes, quadratic_midpoint_witness)

Take c=(a+b)/2, verify a<c<b, and compare 2*A*c+B with the exact secant slope. For A=0 this affine/constant case has many witnesses; the midpoint is not a uniqueness claim. The exact result, retaining the requested order, is (1/2, Yes, 19, 19, Yes, quadratic_midpoint_witness).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r02-mean-value-theorem-0075/)

Mean value theorem · qf-mod41-mvt-polynomial-witness · candidate





## 76. Failed MVT Hypotheses Need Their Own Exact Output Layout

Use boundary mode=degenerate_interval, a=0, b=0, scale=1, offset=0. degenerate_interval declares a≥b; jump_at_zero and cusp_at_zero declare a<0<b and nonzero scale, with the named interior defect. Give (mode, a, b, ordered_endpoints, secant_slope_or_null, theorem_applicability, boundary).

The original mode selects7 fields for degenerate_interval or9 for jump_at_zero/cusp_at_zero. No padding or answer-shape inference is used. null retains the withheld secant value; No withholds theorem applicability under the failed hypotheses.

View answer and explanation(degenerate_interval, 0, 0, No, null, No, a_less_than_b_and_nonzero_secant_denominator_required)

Use the original mode-selected fields, without padding. Degenerate/reversed endpoint mode withholds a secant value as null; even if a>b makes a numerical quotient possible, it fails the ordered-interval theorem scope. A jump fails continuity and differentiability at zero; a cusp is continuous but not differentiable there. No means this failed-hypothesis case cannot invoke MVT, not a general classification of every unrelated function. The exact result, retaining the requested order, is (degenerate_interval, 0, 0, No, null, No, a_less_than_b_and_nonzero_secant_denominator_required).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r02-mean-value-theorem-0076/)

Mean value theorem · qf-mod41-mvt-failure-boundary · candidate





## 77. Failed MVT Hypotheses Need Their Own Exact Output Layout

Use boundary mode=jump_at_zero, a=-2, b=3, scale=2, offset=1. degenerate_interval declares a≥b; jump_at_zero and cusp_at_zero declare a<0<b and nonzero scale, with the named interior defect. Give (mode, a, b, scale, offset, closed_interval_continuity, open_interval_differentiability, theorem_applicability, boundary).

The original mode selects7 fields for degenerate_interval or9 for jump_at_zero/cusp_at_zero. No padding or answer-shape inference is used. null retains the withheld secant value; No withholds theorem applicability under the failed hypotheses.

View answer and explanation(jump_at_zero, -2, 3, 2, 1, No, No, No, endpoint_interval_continuity_fails_at_zero)

Use the original mode-selected fields, without padding. Degenerate/reversed endpoint mode withholds a secant value as null; even if a>b makes a numerical quotient possible, it fails the ordered-interval theorem scope. A jump fails continuity and differentiability at zero; a cusp is continuous but not differentiable there. No means this failed-hypothesis case cannot invoke MVT, not a general classification of every unrelated function. The exact result, retaining the requested order, is (jump_at_zero, -2, 3, 2, 1, No, No, No, endpoint_interval_continuity_fails_at_zero).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r02-mean-value-theorem-0077/)

Mean value theorem · qf-mod41-mvt-failure-boundary · candidate





## 78. Failed MVT Hypotheses Need Their Own Exact Output Layout

Use boundary mode=cusp_at_zero, a=-3, b=4, scale=3, offset=2. degenerate_interval declares a≥b; jump_at_zero and cusp_at_zero declare a<0<b and nonzero scale, with the named interior defect. Give (mode, a, b, scale, offset, closed_interval_continuity, open_interval_differentiability, theorem_applicability, boundary).

The original mode selects7 fields for degenerate_interval or9 for jump_at_zero/cusp_at_zero. No padding or answer-shape inference is used. null retains the withheld secant value; No withholds theorem applicability under the failed hypotheses.

View answer and explanation(cusp_at_zero, -3, 4, 3, 2, Yes, No, No, open_interval_differentiability_fails_at_zero)

Use the original mode-selected fields, without padding. Degenerate/reversed endpoint mode withholds a secant value as null; even if a>b makes a numerical quotient possible, it fails the ordered-interval theorem scope. A jump fails continuity and differentiability at zero; a cusp is continuous but not differentiable there. No means this failed-hypothesis case cannot invoke MVT, not a general classification of every unrelated function. The exact result, retaining the requested order, is (cusp_at_zero, -3, 4, 3, 2, Yes, No, No, open_interval_differentiability_fails_at_zero).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r02-mean-value-theorem-0078/)

Mean value theorem · qf-mod41-mvt-failure-boundary · candidate





## 79. Failed MVT Hypotheses Need Their Own Exact Output Layout

Use boundary mode=degenerate_interval, a=3, b=3, scale=4, offset=-3. degenerate_interval declares a≥b; jump_at_zero and cusp_at_zero declare a<0<b and nonzero scale, with the named interior defect. Give (mode, a, b, ordered_endpoints, secant_slope_or_null, theorem_applicability, boundary).

The original mode selects7 fields for degenerate_interval or9 for jump_at_zero/cusp_at_zero. No padding or answer-shape inference is used. null retains the withheld secant value; No withholds theorem applicability under the failed hypotheses.

View answer and explanation(degenerate_interval, 3, 3, No, null, No, a_less_than_b_and_nonzero_secant_denominator_required)

Use the original mode-selected fields, without padding. Degenerate/reversed endpoint mode withholds a secant value as null; even if a>b makes a numerical quotient possible, it fails the ordered-interval theorem scope. A jump fails continuity and differentiability at zero; a cusp is continuous but not differentiable there. No means this failed-hypothesis case cannot invoke MVT, not a general classification of every unrelated function. The exact result, retaining the requested order, is (degenerate_interval, 3, 3, No, null, No, a_less_than_b_and_nonzero_secant_denominator_required).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r02-mean-value-theorem-0079/)

Mean value theorem · qf-mod41-mvt-failure-boundary · candidate





## 80. Failed MVT Hypotheses Need Their Own Exact Output Layout

Use boundary mode=jump_at_zero, a=-5, b=6, scale=5, offset=4. degenerate_interval declares a≥b; jump_at_zero and cusp_at_zero declare a<0<b and nonzero scale, with the named interior defect. Give (mode, a, b, scale, offset, closed_interval_continuity, open_interval_differentiability, theorem_applicability, boundary).

The original mode selects7 fields for degenerate_interval or9 for jump_at_zero/cusp_at_zero. No padding or answer-shape inference is used. null retains the withheld secant value; No withholds theorem applicability under the failed hypotheses.

View answer and explanation(jump_at_zero, -5, 6, 5, 4, No, No, No, endpoint_interval_continuity_fails_at_zero)

Use the original mode-selected fields, without padding. Degenerate/reversed endpoint mode withholds a secant value as null; even if a>b makes a numerical quotient possible, it fails the ordered-interval theorem scope. A jump fails continuity and differentiability at zero; a cusp is continuous but not differentiable there. No means this failed-hypothesis case cannot invoke MVT, not a general classification of every unrelated function. The exact result, retaining the requested order, is (jump_at_zero, -5, 6, 5, 4, No, No, No, endpoint_interval_continuity_fails_at_zero).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r02-mean-value-theorem-0080/)

Mean value theorem · qf-mod41-mvt-failure-boundary · candidate





## 81. Failed MVT Hypotheses Need Their Own Exact Output Layout

Use boundary mode=cusp_at_zero, a=-6, b=7, scale=6, offset=5. degenerate_interval declares a≥b; jump_at_zero and cusp_at_zero declare a<0<b and nonzero scale, with the named interior defect. Give (mode, a, b, scale, offset, closed_interval_continuity, open_interval_differentiability, theorem_applicability, boundary).

The original mode selects7 fields for degenerate_interval or9 for jump_at_zero/cusp_at_zero. No padding or answer-shape inference is used. null retains the withheld secant value; No withholds theorem applicability under the failed hypotheses.

View answer and explanation(cusp_at_zero, -6, 7, 6, 5, Yes, No, No, open_interval_differentiability_fails_at_zero)

Use the original mode-selected fields, without padding. Degenerate/reversed endpoint mode withholds a secant value as null; even if a>b makes a numerical quotient possible, it fails the ordered-interval theorem scope. A jump fails continuity and differentiability at zero; a cusp is continuous but not differentiable there. No means this failed-hypothesis case cannot invoke MVT, not a general classification of every unrelated function. The exact result, retaining the requested order, is (cusp_at_zero, -6, 7, 6, 5, Yes, No, No, open_interval_differentiability_fails_at_zero).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r02-mean-value-theorem-0081/)

Mean value theorem · qf-mod41-mvt-failure-boundary · candidate





## 82. Failed MVT Hypotheses Need Their Own Exact Output Layout

Use boundary mode=degenerate_interval, a=6, b=6, scale=7, offset=-6. degenerate_interval declares a≥b; jump_at_zero and cusp_at_zero declare a<0<b and nonzero scale, with the named interior defect. Give (mode, a, b, ordered_endpoints, secant_slope_or_null, theorem_applicability, boundary).

The original mode selects7 fields for degenerate_interval or9 for jump_at_zero/cusp_at_zero. No padding or answer-shape inference is used. null retains the withheld secant value; No withholds theorem applicability under the failed hypotheses.

View answer and explanation(degenerate_interval, 6, 6, No, null, No, a_less_than_b_and_nonzero_secant_denominator_required)

Use the original mode-selected fields, without padding. Degenerate/reversed endpoint mode withholds a secant value as null; even if a>b makes a numerical quotient possible, it fails the ordered-interval theorem scope. A jump fails continuity and differentiability at zero; a cusp is continuous but not differentiable there. No means this failed-hypothesis case cannot invoke MVT, not a general classification of every unrelated function. The exact result, retaining the requested order, is (degenerate_interval, 6, 6, No, null, No, a_less_than_b_and_nonzero_secant_denominator_required).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r02-mean-value-theorem-0082/)

Mean value theorem · qf-mod41-mvt-failure-boundary · candidate





## 83. Failed MVT Hypotheses Need Their Own Exact Output Layout

Use boundary mode=jump_at_zero, a=-8, b=9, scale=8, offset=7. degenerate_interval declares a≥b; jump_at_zero and cusp_at_zero declare a<0<b and nonzero scale, with the named interior defect. Give (mode, a, b, scale, offset, closed_interval_continuity, open_interval_differentiability, theorem_applicability, boundary).

The original mode selects7 fields for degenerate_interval or9 for jump_at_zero/cusp_at_zero. No padding or answer-shape inference is used. null retains the withheld secant value; No withholds theorem applicability under the failed hypotheses.

View answer and explanation(jump_at_zero, -8, 9, 8, 7, No, No, No, endpoint_interval_continuity_fails_at_zero)

Use the original mode-selected fields, without padding. Degenerate/reversed endpoint mode withholds a secant value as null; even if a>b makes a numerical quotient possible, it fails the ordered-interval theorem scope. A jump fails continuity and differentiability at zero; a cusp is continuous but not differentiable there. No means this failed-hypothesis case cannot invoke MVT, not a general classification of every unrelated function. The exact result, retaining the requested order, is (jump_at_zero, -8, 9, 8, 7, No, No, No, endpoint_interval_continuity_fails_at_zero).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r02-mean-value-theorem-0083/)

Mean value theorem · qf-mod41-mvt-failure-boundary · candidate





## 84. Failed MVT Hypotheses Need Their Own Exact Output Layout

Use boundary mode=cusp_at_zero, a=-9, b=10, scale=9, offset=8. degenerate_interval declares a≥b; jump_at_zero and cusp_at_zero declare a<0<b and nonzero scale, with the named interior defect. Give (mode, a, b, scale, offset, closed_interval_continuity, open_interval_differentiability, theorem_applicability, boundary).

The original mode selects7 fields for degenerate_interval or9 for jump_at_zero/cusp_at_zero. No padding or answer-shape inference is used. null retains the withheld secant value; No withholds theorem applicability under the failed hypotheses.

View answer and explanation(cusp_at_zero, -9, 10, 9, 8, Yes, No, No, open_interval_differentiability_fails_at_zero)

Use the original mode-selected fields, without padding. Degenerate/reversed endpoint mode withholds a secant value as null; even if a>b makes a numerical quotient possible, it fails the ordered-interval theorem scope. A jump fails continuity and differentiability at zero; a cusp is continuous but not differentiable there. No means this failed-hypothesis case cannot invoke MVT, not a general classification of every unrelated function. The exact result, retaining the requested order, is (cusp_at_zero, -9, 10, 9, 8, Yes, No, No, open_interval_differentiability_fails_at_zero).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r02-mean-value-theorem-0084/)

Mean value theorem · qf-mod41-mvt-failure-boundary · candidate





## 85. Failed MVT Hypotheses Need Their Own Exact Output Layout

Use boundary mode=degenerate_interval, a=9, b=9, scale=10, offset=-9. degenerate_interval declares a≥b; jump_at_zero and cusp_at_zero declare a<0<b and nonzero scale, with the named interior defect. Give (mode, a, b, ordered_endpoints, secant_slope_or_null, theorem_applicability, boundary).

The original mode selects7 fields for degenerate_interval or9 for jump_at_zero/cusp_at_zero. No padding or answer-shape inference is used. null retains the withheld secant value; No withholds theorem applicability under the failed hypotheses.

View answer and explanation(degenerate_interval, 9, 9, No, null, No, a_less_than_b_and_nonzero_secant_denominator_required)

Use the original mode-selected fields, without padding. Degenerate/reversed endpoint mode withholds a secant value as null; even if a>b makes a numerical quotient possible, it fails the ordered-interval theorem scope. A jump fails continuity and differentiability at zero; a cusp is continuous but not differentiable there. No means this failed-hypothesis case cannot invoke MVT, not a general classification of every unrelated function. The exact result, retaining the requested order, is (degenerate_interval, 9, 9, No, null, No, a_less_than_b_and_nonzero_secant_denominator_required).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r02-mean-value-theorem-0085/)

Mean value theorem · qf-mod41-mvt-failure-boundary · candidate





## 86. Failed MVT Hypotheses Need Their Own Exact Output Layout

Use boundary mode=jump_at_zero, a=-11, b=12, scale=11, offset=10. degenerate_interval declares a≥b; jump_at_zero and cusp_at_zero declare a<0<b and nonzero scale, with the named interior defect. Give (mode, a, b, scale, offset, closed_interval_continuity, open_interval_differentiability, theorem_applicability, boundary).

The original mode selects7 fields for degenerate_interval or9 for jump_at_zero/cusp_at_zero. No padding or answer-shape inference is used. null retains the withheld secant value; No withholds theorem applicability under the failed hypotheses.

View answer and explanation(jump_at_zero, -11, 12, 11, 10, No, No, No, endpoint_interval_continuity_fails_at_zero)

Use the original mode-selected fields, without padding. Degenerate/reversed endpoint mode withholds a secant value as null; even if a>b makes a numerical quotient possible, it fails the ordered-interval theorem scope. A jump fails continuity and differentiability at zero; a cusp is continuous but not differentiable there. No means this failed-hypothesis case cannot invoke MVT, not a general classification of every unrelated function. The exact result, retaining the requested order, is (jump_at_zero, -11, 12, 11, 10, No, No, No, endpoint_interval_continuity_fails_at_zero).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r02-mean-value-theorem-0086/)

Mean value theorem · qf-mod41-mvt-failure-boundary · candidate





## 87. Failed MVT Hypotheses Need Their Own Exact Output Layout

Use boundary mode=cusp_at_zero, a=-12, b=13, scale=12, offset=11. degenerate_interval declares a≥b; jump_at_zero and cusp_at_zero declare a<0<b and nonzero scale, with the named interior defect. Give (mode, a, b, scale, offset, closed_interval_continuity, open_interval_differentiability, theorem_applicability, boundary).

The original mode selects7 fields for degenerate_interval or9 for jump_at_zero/cusp_at_zero. No padding or answer-shape inference is used. null retains the withheld secant value; No withholds theorem applicability under the failed hypotheses.

View answer and explanation(cusp_at_zero, -12, 13, 12, 11, Yes, No, No, open_interval_differentiability_fails_at_zero)

Use the original mode-selected fields, without padding. Degenerate/reversed endpoint mode withholds a secant value as null; even if a>b makes a numerical quotient possible, it fails the ordered-interval theorem scope. A jump fails continuity and differentiability at zero; a cusp is continuous but not differentiable there. No means this failed-hypothesis case cannot invoke MVT, not a general classification of every unrelated function. The exact result, retaining the requested order, is (cusp_at_zero, -12, 13, 12, 11, Yes, No, No, open_interval_differentiability_fails_at_zero).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r02-mean-value-theorem-0087/)

Mean value theorem · qf-mod41-mvt-failure-boundary · candidate





## 88. Failed MVT Hypotheses Need Their Own Exact Output Layout

Use boundary mode=degenerate_interval, a=12, b=12, scale=13, offset=-12. degenerate_interval declares a≥b; jump_at_zero and cusp_at_zero declare a<0<b and nonzero scale, with the named interior defect. Give (mode, a, b, ordered_endpoints, secant_slope_or_null, theorem_applicability, boundary).

The original mode selects7 fields for degenerate_interval or9 for jump_at_zero/cusp_at_zero. No padding or answer-shape inference is used. null retains the withheld secant value; No withholds theorem applicability under the failed hypotheses.

View answer and explanation(degenerate_interval, 12, 12, No, null, No, a_less_than_b_and_nonzero_secant_denominator_required)

Use the original mode-selected fields, without padding. Degenerate/reversed endpoint mode withholds a secant value as null; even if a>b makes a numerical quotient possible, it fails the ordered-interval theorem scope. A jump fails continuity and differentiability at zero; a cusp is continuous but not differentiable there. No means this failed-hypothesis case cannot invoke MVT, not a general classification of every unrelated function. The exact result, retaining the requested order, is (degenerate_interval, 12, 12, No, null, No, a_less_than_b_and_nonzero_secant_denominator_required).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r02-mean-value-theorem-0088/)

Mean value theorem · qf-mod41-mvt-failure-boundary · candidate





## 89. Failed MVT Hypotheses Need Their Own Exact Output Layout

Use boundary mode=jump_at_zero, a=-14, b=15, scale=14, offset=13. degenerate_interval declares a≥b; jump_at_zero and cusp_at_zero declare a<0<b and nonzero scale, with the named interior defect. Give (mode, a, b, scale, offset, closed_interval_continuity, open_interval_differentiability, theorem_applicability, boundary).

The original mode selects7 fields for degenerate_interval or9 for jump_at_zero/cusp_at_zero. No padding or answer-shape inference is used. null retains the withheld secant value; No withholds theorem applicability under the failed hypotheses.

View answer and explanation(jump_at_zero, -14, 15, 14, 13, No, No, No, endpoint_interval_continuity_fails_at_zero)

Use the original mode-selected fields, without padding. Degenerate/reversed endpoint mode withholds a secant value as null; even if a>b makes a numerical quotient possible, it fails the ordered-interval theorem scope. A jump fails continuity and differentiability at zero; a cusp is continuous but not differentiable there. No means this failed-hypothesis case cannot invoke MVT, not a general classification of every unrelated function. The exact result, retaining the requested order, is (jump_at_zero, -14, 15, 14, 13, No, No, No, endpoint_interval_continuity_fails_at_zero).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r02-mean-value-theorem-0089/)

Mean value theorem · qf-mod41-mvt-failure-boundary · candidate





## 90. Failed MVT Hypotheses Need Their Own Exact Output Layout

Use boundary mode=cusp_at_zero, a=-15, b=16, scale=15, offset=14. degenerate_interval declares a≥b; jump_at_zero and cusp_at_zero declare a<0<b and nonzero scale, with the named interior defect. Give (mode, a, b, scale, offset, closed_interval_continuity, open_interval_differentiability, theorem_applicability, boundary).

The original mode selects7 fields for degenerate_interval or9 for jump_at_zero/cusp_at_zero. No padding or answer-shape inference is used. null retains the withheld secant value; No withholds theorem applicability under the failed hypotheses.

View answer and explanation(cusp_at_zero, -15, 16, 15, 14, Yes, No, No, open_interval_differentiability_fails_at_zero)

Use the original mode-selected fields, without padding. Degenerate/reversed endpoint mode withholds a secant value as null; even if a>b makes a numerical quotient possible, it fails the ordered-interval theorem scope. A jump fails continuity and differentiability at zero; a cusp is continuous but not differentiable there. No means this failed-hypothesis case cannot invoke MVT, not a general classification of every unrelated function. The exact result, retaining the requested order, is (cusp_at_zero, -15, 16, 15, 14, Yes, No, No, open_interval_differentiability_fails_at_zero).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r02-mean-value-theorem-0090/)

Mean value theorem · qf-mod41-mvt-failure-boundary · candidate





## 91. Failed MVT Hypotheses Need Their Own Exact Output Layout

Use boundary mode=degenerate_interval, a=15, b=15, scale=16, offset=-15. degenerate_interval declares a≥b; jump_at_zero and cusp_at_zero declare a<0<b and nonzero scale, with the named interior defect. Give (mode, a, b, ordered_endpoints, secant_slope_or_null, theorem_applicability, boundary).

The original mode selects7 fields for degenerate_interval or9 for jump_at_zero/cusp_at_zero. No padding or answer-shape inference is used. null retains the withheld secant value; No withholds theorem applicability under the failed hypotheses.

View answer and explanation(degenerate_interval, 15, 15, No, null, No, a_less_than_b_and_nonzero_secant_denominator_required)

Use the original mode-selected fields, without padding. Degenerate/reversed endpoint mode withholds a secant value as null; even if a>b makes a numerical quotient possible, it fails the ordered-interval theorem scope. A jump fails continuity and differentiability at zero; a cusp is continuous but not differentiable there. No means this failed-hypothesis case cannot invoke MVT, not a general classification of every unrelated function. The exact result, retaining the requested order, is (degenerate_interval, 15, 15, No, null, No, a_less_than_b_and_nonzero_secant_denominator_required).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r02-mean-value-theorem-0091/)

Mean value theorem · qf-mod41-mvt-failure-boundary · candidate





## 92. Failed MVT Hypotheses Need Their Own Exact Output Layout

Use boundary mode=jump_at_zero, a=-17, b=18, scale=17, offset=16. degenerate_interval declares a≥b; jump_at_zero and cusp_at_zero declare a<0<b and nonzero scale, with the named interior defect. Give (mode, a, b, scale, offset, closed_interval_continuity, open_interval_differentiability, theorem_applicability, boundary).

The original mode selects7 fields for degenerate_interval or9 for jump_at_zero/cusp_at_zero. No padding or answer-shape inference is used. null retains the withheld secant value; No withholds theorem applicability under the failed hypotheses.

View answer and explanation(jump_at_zero, -17, 18, 17, 16, No, No, No, endpoint_interval_continuity_fails_at_zero)

Use the original mode-selected fields, without padding. Degenerate/reversed endpoint mode withholds a secant value as null; even if a>b makes a numerical quotient possible, it fails the ordered-interval theorem scope. A jump fails continuity and differentiability at zero; a cusp is continuous but not differentiable there. No means this failed-hypothesis case cannot invoke MVT, not a general classification of every unrelated function. The exact result, retaining the requested order, is (jump_at_zero, -17, 18, 17, 16, No, No, No, endpoint_interval_continuity_fails_at_zero).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r02-mean-value-theorem-0092/)

Mean value theorem · qf-mod41-mvt-failure-boundary · candidate





## 93. Failed MVT Hypotheses Need Their Own Exact Output Layout

Use boundary mode=cusp_at_zero, a=-18, b=19, scale=18, offset=17. degenerate_interval declares a≥b; jump_at_zero and cusp_at_zero declare a<0<b and nonzero scale, with the named interior defect. Give (mode, a, b, scale, offset, closed_interval_continuity, open_interval_differentiability, theorem_applicability, boundary).

The original mode selects7 fields for degenerate_interval or9 for jump_at_zero/cusp_at_zero. No padding or answer-shape inference is used. null retains the withheld secant value; No withholds theorem applicability under the failed hypotheses.

View answer and explanation(cusp_at_zero, -18, 19, 18, 17, Yes, No, No, open_interval_differentiability_fails_at_zero)

Use the original mode-selected fields, without padding. Degenerate/reversed endpoint mode withholds a secant value as null; even if a>b makes a numerical quotient possible, it fails the ordered-interval theorem scope. A jump fails continuity and differentiability at zero; a cusp is continuous but not differentiable there. No means this failed-hypothesis case cannot invoke MVT, not a general classification of every unrelated function. The exact result, retaining the requested order, is (cusp_at_zero, -18, 19, 18, 17, Yes, No, No, open_interval_differentiability_fails_at_zero).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r02-mean-value-theorem-0093/)

Mean value theorem · qf-mod41-mvt-failure-boundary · candidate





## 94. Failed MVT Hypotheses Need Their Own Exact Output Layout

Use boundary mode=degenerate_interval, a=18, b=18, scale=19, offset=-18. degenerate_interval declares a≥b; jump_at_zero and cusp_at_zero declare a<0<b and nonzero scale, with the named interior defect. Give (mode, a, b, ordered_endpoints, secant_slope_or_null, theorem_applicability, boundary).

The original mode selects7 fields for degenerate_interval or9 for jump_at_zero/cusp_at_zero. No padding or answer-shape inference is used. null retains the withheld secant value; No withholds theorem applicability under the failed hypotheses.

View answer and explanation(degenerate_interval, 18, 18, No, null, No, a_less_than_b_and_nonzero_secant_denominator_required)

Use the original mode-selected fields, without padding. Degenerate/reversed endpoint mode withholds a secant value as null; even if a>b makes a numerical quotient possible, it fails the ordered-interval theorem scope. A jump fails continuity and differentiability at zero; a cusp is continuous but not differentiable there. No means this failed-hypothesis case cannot invoke MVT, not a general classification of every unrelated function. The exact result, retaining the requested order, is (degenerate_interval, 18, 18, No, null, No, a_less_than_b_and_nonzero_secant_denominator_required).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r02-mean-value-theorem-0094/)

Mean value theorem · qf-mod41-mvt-failure-boundary · candidate





## 95. Failed MVT Hypotheses Need Their Own Exact Output Layout

Use boundary mode=jump_at_zero, a=-20, b=21, scale=20, offset=19. degenerate_interval declares a≥b; jump_at_zero and cusp_at_zero declare a<0<b and nonzero scale, with the named interior defect. Give (mode, a, b, scale, offset, closed_interval_continuity, open_interval_differentiability, theorem_applicability, boundary).

The original mode selects7 fields for degenerate_interval or9 for jump_at_zero/cusp_at_zero. No padding or answer-shape inference is used. null retains the withheld secant value; No withholds theorem applicability under the failed hypotheses.

View answer and explanation(jump_at_zero, -20, 21, 20, 19, No, No, No, endpoint_interval_continuity_fails_at_zero)

Use the original mode-selected fields, without padding. Degenerate/reversed endpoint mode withholds a secant value as null; even if a>b makes a numerical quotient possible, it fails the ordered-interval theorem scope. A jump fails continuity and differentiability at zero; a cusp is continuous but not differentiable there. No means this failed-hypothesis case cannot invoke MVT, not a general classification of every unrelated function. The exact result, retaining the requested order, is (jump_at_zero, -20, 21, 20, 19, No, No, No, endpoint_interval_continuity_fails_at_zero).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r02-mean-value-theorem-0095/)

Mean value theorem · qf-mod41-mvt-failure-boundary · candidate





## 96. Failed MVT Hypotheses Need Their Own Exact Output Layout

Use boundary mode=cusp_at_zero, a=-21, b=22, scale=21, offset=20. degenerate_interval declares a≥b; jump_at_zero and cusp_at_zero declare a<0<b and nonzero scale, with the named interior defect. Give (mode, a, b, scale, offset, closed_interval_continuity, open_interval_differentiability, theorem_applicability, boundary).

The original mode selects7 fields for degenerate_interval or9 for jump_at_zero/cusp_at_zero. No padding or answer-shape inference is used. null retains the withheld secant value; No withholds theorem applicability under the failed hypotheses.

View answer and explanation(cusp_at_zero, -21, 22, 21, 20, Yes, No, No, open_interval_differentiability_fails_at_zero)

Use the original mode-selected fields, without padding. Degenerate/reversed endpoint mode withholds a secant value as null; even if a>b makes a numerical quotient possible, it fails the ordered-interval theorem scope. A jump fails continuity and differentiability at zero; a cusp is continuous but not differentiable there. No means this failed-hypothesis case cannot invoke MVT, not a general classification of every unrelated function. The exact result, retaining the requested order, is (cusp_at_zero, -21, 22, 21, 20, Yes, No, No, open_interval_differentiability_fails_at_zero).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r02-mean-value-theorem-0096/)

Mean value theorem · qf-mod41-mvt-failure-boundary · candidate





## 97. Failed MVT Hypotheses Need Their Own Exact Output Layout

Use boundary mode=degenerate_interval, a=21, b=21, scale=22, offset=-21. degenerate_interval declares a≥b; jump_at_zero and cusp_at_zero declare a<0<b and nonzero scale, with the named interior defect. Give (mode, a, b, ordered_endpoints, secant_slope_or_null, theorem_applicability, boundary).

The original mode selects7 fields for degenerate_interval or9 for jump_at_zero/cusp_at_zero. No padding or answer-shape inference is used. null retains the withheld secant value; No withholds theorem applicability under the failed hypotheses.

View answer and explanation(degenerate_interval, 21, 21, No, null, No, a_less_than_b_and_nonzero_secant_denominator_required)

Use the original mode-selected fields, without padding. Degenerate/reversed endpoint mode withholds a secant value as null; even if a>b makes a numerical quotient possible, it fails the ordered-interval theorem scope. A jump fails continuity and differentiability at zero; a cusp is continuous but not differentiable there. No means this failed-hypothesis case cannot invoke MVT, not a general classification of every unrelated function. The exact result, retaining the requested order, is (degenerate_interval, 21, 21, No, null, No, a_less_than_b_and_nonzero_secant_denominator_required).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r02-mean-value-theorem-0097/)

Mean value theorem · qf-mod41-mvt-failure-boundary · candidate





## 98. Failed MVT Hypotheses Need Their Own Exact Output Layout

Use boundary mode=jump_at_zero, a=-23, b=24, scale=23, offset=22. degenerate_interval declares a≥b; jump_at_zero and cusp_at_zero declare a<0<b and nonzero scale, with the named interior defect. Give (mode, a, b, scale, offset, closed_interval_continuity, open_interval_differentiability, theorem_applicability, boundary).

The original mode selects7 fields for degenerate_interval or9 for jump_at_zero/cusp_at_zero. No padding or answer-shape inference is used. null retains the withheld secant value; No withholds theorem applicability under the failed hypotheses.

View answer and explanation(jump_at_zero, -23, 24, 23, 22, No, No, No, endpoint_interval_continuity_fails_at_zero)

Use the original mode-selected fields, without padding. Degenerate/reversed endpoint mode withholds a secant value as null; even if a>b makes a numerical quotient possible, it fails the ordered-interval theorem scope. A jump fails continuity and differentiability at zero; a cusp is continuous but not differentiable there. No means this failed-hypothesis case cannot invoke MVT, not a general classification of every unrelated function. The exact result, retaining the requested order, is (jump_at_zero, -23, 24, 23, 22, No, No, No, endpoint_interval_continuity_fails_at_zero).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r02-mean-value-theorem-0098/)

Mean value theorem · qf-mod41-mvt-failure-boundary · candidate





## 99. Failed MVT Hypotheses Need Their Own Exact Output Layout

Use boundary mode=cusp_at_zero, a=-24, b=25, scale=24, offset=23. degenerate_interval declares a≥b; jump_at_zero and cusp_at_zero declare a<0<b and nonzero scale, with the named interior defect. Give (mode, a, b, scale, offset, closed_interval_continuity, open_interval_differentiability, theorem_applicability, boundary).

The original mode selects7 fields for degenerate_interval or9 for jump_at_zero/cusp_at_zero. No padding or answer-shape inference is used. null retains the withheld secant value; No withholds theorem applicability under the failed hypotheses.

View answer and explanation(cusp_at_zero, -24, 25, 24, 23, Yes, No, No, open_interval_differentiability_fails_at_zero)

Use the original mode-selected fields, without padding. Degenerate/reversed endpoint mode withholds a secant value as null; even if a>b makes a numerical quotient possible, it fails the ordered-interval theorem scope. A jump fails continuity and differentiability at zero; a cusp is continuous but not differentiable there. No means this failed-hypothesis case cannot invoke MVT, not a general classification of every unrelated function. The exact result, retaining the requested order, is (cusp_at_zero, -24, 25, 24, 23, Yes, No, No, open_interval_differentiability_fails_at_zero).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r02-mean-value-theorem-0099/)

Mean value theorem · qf-mod41-mvt-failure-boundary · candidate





## 100. Failed MVT Hypotheses Need Their Own Exact Output Layout

Use boundary mode=degenerate_interval, a=24, b=24, scale=25, offset=-24. degenerate_interval declares a≥b; jump_at_zero and cusp_at_zero declare a<0<b and nonzero scale, with the named interior defect. Give (mode, a, b, ordered_endpoints, secant_slope_or_null, theorem_applicability, boundary).

The original mode selects7 fields for degenerate_interval or9 for jump_at_zero/cusp_at_zero. No padding or answer-shape inference is used. null retains the withheld secant value; No withholds theorem applicability under the failed hypotheses.

View answer and explanation(degenerate_interval, 24, 24, No, null, No, a_less_than_b_and_nonzero_secant_denominator_required)

Use the original mode-selected fields, without padding. Degenerate/reversed endpoint mode withholds a secant value as null; even if a>b makes a numerical quotient possible, it fails the ordered-interval theorem scope. A jump fails continuity and differentiability at zero; a cusp is continuous but not differentiable there. No means this failed-hypothesis case cannot invoke MVT, not a general classification of every unrelated function. The exact result, retaining the requested order, is (degenerate_interval, 24, 24, No, null, No, a_less_than_b_and_nonzero_secant_denominator_required).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r02-mean-value-theorem-0100/)

Mean value theorem · qf-mod41-mvt-failure-boundary · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261110-r02-mean-value-theorem-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261014-advance-20261110-r02-mean-value-theorem-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261110-r02-mean-value-theorem-foundations/validation-report.json)