# Inclusive IVT Witnesses and Mode-Specific Continuity Boundaries

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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-20261015-advance-20261116-r02-intermediate-value-theorem-foundations · 2026-10-15 · 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. Polynomial Continuity and Inclusive Endpoint Bracketing Are Separate Obligations

Coefficients=[12, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=-13<b=-11 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Give (coefficients, a, b, value_at_a, value_at_b, target, polynomial_continuous, inclusive_bracket, certificate).

View answer and explanation([12, 1], -13, -11, -1, 1, 0, Yes, Yes, polynomial_continuity_plus_inclusive_endpoint_bracket)

Evaluate both endpoints exactly and test target-between-values in either orientation, including equality at an endpoint. A target outside the bracket does not satisfy this IVT premise; finite sample agreement does not establish continuity of an arbitrary function. The exact result, retaining the requested order, is ([12, 1], -13, -11, -1, 1, 0, Yes, Yes, polynomial_continuity_plus_inclusive_endpoint_bracket).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r02-intermediate-value-theorem-0001/)

Intermediate value theorem · qf-mod47-ivt-hypothesis · candidate





## 2. Polynomial Continuity and Inclusive Endpoint Bracketing Are Separate Obligations

Coefficients=[11, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=-12<b=-10 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Give (coefficients, a, b, value_at_a, value_at_b, target, polynomial_continuous, inclusive_bracket, certificate).

View answer and explanation([11, 1], -12, -10, -1, 1, 0, Yes, Yes, polynomial_continuity_plus_inclusive_endpoint_bracket)

Evaluate both endpoints exactly and test target-between-values in either orientation, including equality at an endpoint. A target outside the bracket does not satisfy this IVT premise; finite sample agreement does not establish continuity of an arbitrary function. The exact result, retaining the requested order, is ([11, 1], -12, -10, -1, 1, 0, Yes, Yes, polynomial_continuity_plus_inclusive_endpoint_bracket).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r02-intermediate-value-theorem-0002/)

Intermediate value theorem · qf-mod47-ivt-hypothesis · candidate





## 3. Polynomial Continuity and Inclusive Endpoint Bracketing Are Separate Obligations

Coefficients=[10, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=-11<b=-9 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Give (coefficients, a, b, value_at_a, value_at_b, target, polynomial_continuous, inclusive_bracket, certificate).

View answer and explanation([10, 1], -11, -9, -1, 1, 0, Yes, Yes, polynomial_continuity_plus_inclusive_endpoint_bracket)

Evaluate both endpoints exactly and test target-between-values in either orientation, including equality at an endpoint. A target outside the bracket does not satisfy this IVT premise; finite sample agreement does not establish continuity of an arbitrary function. The exact result, retaining the requested order, is ([10, 1], -11, -9, -1, 1, 0, Yes, Yes, polynomial_continuity_plus_inclusive_endpoint_bracket).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r02-intermediate-value-theorem-0003/)

Intermediate value theorem · qf-mod47-ivt-hypothesis · candidate





## 4. Polynomial Continuity and Inclusive Endpoint Bracketing Are Separate Obligations

Coefficients=[9, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=-10<b=-8 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Give (coefficients, a, b, value_at_a, value_at_b, target, polynomial_continuous, inclusive_bracket, certificate).

View answer and explanation([9, 1], -10, -8, -1, 1, 0, Yes, Yes, polynomial_continuity_plus_inclusive_endpoint_bracket)

Evaluate both endpoints exactly and test target-between-values in either orientation, including equality at an endpoint. A target outside the bracket does not satisfy this IVT premise; finite sample agreement does not establish continuity of an arbitrary function. The exact result, retaining the requested order, is ([9, 1], -10, -8, -1, 1, 0, Yes, Yes, polynomial_continuity_plus_inclusive_endpoint_bracket).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r02-intermediate-value-theorem-0004/)

Intermediate value theorem · qf-mod47-ivt-hypothesis · candidate





## 5. Polynomial Continuity and Inclusive Endpoint Bracketing Are Separate Obligations

Coefficients=[8, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=-9<b=-7 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Give (coefficients, a, b, value_at_a, value_at_b, target, polynomial_continuous, inclusive_bracket, certificate).

View answer and explanation([8, 1], -9, -7, -1, 1, 0, Yes, Yes, polynomial_continuity_plus_inclusive_endpoint_bracket)

Evaluate both endpoints exactly and test target-between-values in either orientation, including equality at an endpoint. A target outside the bracket does not satisfy this IVT premise; finite sample agreement does not establish continuity of an arbitrary function. The exact result, retaining the requested order, is ([8, 1], -9, -7, -1, 1, 0, Yes, Yes, polynomial_continuity_plus_inclusive_endpoint_bracket).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r02-intermediate-value-theorem-0005/)

Intermediate value theorem · qf-mod47-ivt-hypothesis · candidate





## 6. Polynomial Continuity and Inclusive Endpoint Bracketing Are Separate Obligations

Coefficients=[7, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=-8<b=-6 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Give (coefficients, a, b, value_at_a, value_at_b, target, polynomial_continuous, inclusive_bracket, certificate).

View answer and explanation([7, 1], -8, -6, -1, 1, 0, Yes, Yes, polynomial_continuity_plus_inclusive_endpoint_bracket)

Evaluate both endpoints exactly and test target-between-values in either orientation, including equality at an endpoint. A target outside the bracket does not satisfy this IVT premise; finite sample agreement does not establish continuity of an arbitrary function. The exact result, retaining the requested order, is ([7, 1], -8, -6, -1, 1, 0, Yes, Yes, polynomial_continuity_plus_inclusive_endpoint_bracket).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r02-intermediate-value-theorem-0006/)

Intermediate value theorem · qf-mod47-ivt-hypothesis · candidate





## 7. Polynomial Continuity and Inclusive Endpoint Bracketing Are Separate Obligations

Coefficients=[6, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=-7<b=-5 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Give (coefficients, a, b, value_at_a, value_at_b, target, polynomial_continuous, inclusive_bracket, certificate).

View answer and explanation([6, 1], -7, -5, -1, 1, 0, Yes, Yes, polynomial_continuity_plus_inclusive_endpoint_bracket)

Evaluate both endpoints exactly and test target-between-values in either orientation, including equality at an endpoint. A target outside the bracket does not satisfy this IVT premise; finite sample agreement does not establish continuity of an arbitrary function. The exact result, retaining the requested order, is ([6, 1], -7, -5, -1, 1, 0, Yes, Yes, polynomial_continuity_plus_inclusive_endpoint_bracket).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r02-intermediate-value-theorem-0007/)

Intermediate value theorem · qf-mod47-ivt-hypothesis · candidate





## 8. Polynomial Continuity and Inclusive Endpoint Bracketing Are Separate Obligations

Coefficients=[5, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=-6<b=-4 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Give (coefficients, a, b, value_at_a, value_at_b, target, polynomial_continuous, inclusive_bracket, certificate).

View answer and explanation([5, 1], -6, -4, -1, 1, 0, Yes, Yes, polynomial_continuity_plus_inclusive_endpoint_bracket)

Evaluate both endpoints exactly and test target-between-values in either orientation, including equality at an endpoint. A target outside the bracket does not satisfy this IVT premise; finite sample agreement does not establish continuity of an arbitrary function. The exact result, retaining the requested order, is ([5, 1], -6, -4, -1, 1, 0, Yes, Yes, polynomial_continuity_plus_inclusive_endpoint_bracket).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r02-intermediate-value-theorem-0008/)

Intermediate value theorem · qf-mod47-ivt-hypothesis · candidate





## 9. Polynomial Continuity and Inclusive Endpoint Bracketing Are Separate Obligations

Coefficients=[4, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=-5<b=-3 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Give (coefficients, a, b, value_at_a, value_at_b, target, polynomial_continuous, inclusive_bracket, certificate).

View answer and explanation([4, 1], -5, -3, -1, 1, 0, Yes, Yes, polynomial_continuity_plus_inclusive_endpoint_bracket)

Evaluate both endpoints exactly and test target-between-values in either orientation, including equality at an endpoint. A target outside the bracket does not satisfy this IVT premise; finite sample agreement does not establish continuity of an arbitrary function. The exact result, retaining the requested order, is ([4, 1], -5, -3, -1, 1, 0, Yes, Yes, polynomial_continuity_plus_inclusive_endpoint_bracket).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r02-intermediate-value-theorem-0009/)

Intermediate value theorem · qf-mod47-ivt-hypothesis · candidate





## 10. Polynomial Continuity and Inclusive Endpoint Bracketing Are Separate Obligations

Coefficients=[3, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=-4<b=-2 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Give (coefficients, a, b, value_at_a, value_at_b, target, polynomial_continuous, inclusive_bracket, certificate).

View answer and explanation([3, 1], -4, -2, -1, 1, 0, Yes, Yes, polynomial_continuity_plus_inclusive_endpoint_bracket)

Evaluate both endpoints exactly and test target-between-values in either orientation, including equality at an endpoint. A target outside the bracket does not satisfy this IVT premise; finite sample agreement does not establish continuity of an arbitrary function. The exact result, retaining the requested order, is ([3, 1], -4, -2, -1, 1, 0, Yes, Yes, polynomial_continuity_plus_inclusive_endpoint_bracket).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r02-intermediate-value-theorem-0010/)

Intermediate value theorem · qf-mod47-ivt-hypothesis · candidate





## 11. Polynomial Continuity and Inclusive Endpoint Bracketing Are Separate Obligations

Coefficients=[2, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=-3<b=-1 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Give (coefficients, a, b, value_at_a, value_at_b, target, polynomial_continuous, inclusive_bracket, certificate).

View answer and explanation([2, 1], -3, -1, -1, 1, 0, Yes, Yes, polynomial_continuity_plus_inclusive_endpoint_bracket)

Evaluate both endpoints exactly and test target-between-values in either orientation, including equality at an endpoint. A target outside the bracket does not satisfy this IVT premise; finite sample agreement does not establish continuity of an arbitrary function. The exact result, retaining the requested order, is ([2, 1], -3, -1, -1, 1, 0, Yes, Yes, polynomial_continuity_plus_inclusive_endpoint_bracket).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r02-intermediate-value-theorem-0011/)

Intermediate value theorem · qf-mod47-ivt-hypothesis · candidate





## 12. Polynomial Continuity and Inclusive Endpoint Bracketing Are Separate Obligations

Coefficients=[1, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=-2<b=0 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Give (coefficients, a, b, value_at_a, value_at_b, target, polynomial_continuous, inclusive_bracket, certificate).

View answer and explanation([1, 1], -2, 0, -1, 1, 0, Yes, Yes, polynomial_continuity_plus_inclusive_endpoint_bracket)

Evaluate both endpoints exactly and test target-between-values in either orientation, including equality at an endpoint. A target outside the bracket does not satisfy this IVT premise; finite sample agreement does not establish continuity of an arbitrary function. The exact result, retaining the requested order, is ([1, 1], -2, 0, -1, 1, 0, Yes, Yes, polynomial_continuity_plus_inclusive_endpoint_bracket).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r02-intermediate-value-theorem-0012/)

Intermediate value theorem · qf-mod47-ivt-hypothesis · candidate





## 13. Polynomial Continuity and Inclusive Endpoint Bracketing Are Separate Obligations

Coefficients=[0, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=-1<b=1 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Give (coefficients, a, b, value_at_a, value_at_b, target, polynomial_continuous, inclusive_bracket, certificate).

View answer and explanation([0, 1], -1, 1, -1, 1, 0, Yes, Yes, polynomial_continuity_plus_inclusive_endpoint_bracket)

Evaluate both endpoints exactly and test target-between-values in either orientation, including equality at an endpoint. A target outside the bracket does not satisfy this IVT premise; finite sample agreement does not establish continuity of an arbitrary function. The exact result, retaining the requested order, is ([0, 1], -1, 1, -1, 1, 0, Yes, Yes, polynomial_continuity_plus_inclusive_endpoint_bracket).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r02-intermediate-value-theorem-0013/)

Intermediate value theorem · qf-mod47-ivt-hypothesis · candidate





## 14. Polynomial Continuity and Inclusive Endpoint Bracketing Are Separate Obligations

Coefficients=[-1, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=0<b=2 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Give (coefficients, a, b, value_at_a, value_at_b, target, polynomial_continuous, inclusive_bracket, certificate).

View answer and explanation([-1, 1], 0, 2, -1, 1, 0, Yes, Yes, polynomial_continuity_plus_inclusive_endpoint_bracket)

Evaluate both endpoints exactly and test target-between-values in either orientation, including equality at an endpoint. A target outside the bracket does not satisfy this IVT premise; finite sample agreement does not establish continuity of an arbitrary function. The exact result, retaining the requested order, is ([-1, 1], 0, 2, -1, 1, 0, Yes, Yes, polynomial_continuity_plus_inclusive_endpoint_bracket).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r02-intermediate-value-theorem-0014/)

Intermediate value theorem · qf-mod47-ivt-hypothesis · candidate





## 15. Polynomial Continuity and Inclusive Endpoint Bracketing Are Separate Obligations

Coefficients=[-2, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=1<b=3 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Give (coefficients, a, b, value_at_a, value_at_b, target, polynomial_continuous, inclusive_bracket, certificate).

View answer and explanation([-2, 1], 1, 3, -1, 1, 0, Yes, Yes, polynomial_continuity_plus_inclusive_endpoint_bracket)

Evaluate both endpoints exactly and test target-between-values in either orientation, including equality at an endpoint. A target outside the bracket does not satisfy this IVT premise; finite sample agreement does not establish continuity of an arbitrary function. The exact result, retaining the requested order, is ([-2, 1], 1, 3, -1, 1, 0, Yes, Yes, polynomial_continuity_plus_inclusive_endpoint_bracket).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r02-intermediate-value-theorem-0015/)

Intermediate value theorem · qf-mod47-ivt-hypothesis · candidate





## 16. Polynomial Continuity and Inclusive Endpoint Bracketing Are Separate Obligations

Coefficients=[-3, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=2<b=4 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Give (coefficients, a, b, value_at_a, value_at_b, target, polynomial_continuous, inclusive_bracket, certificate).

View answer and explanation([-3, 1], 2, 4, -1, 1, 0, Yes, Yes, polynomial_continuity_plus_inclusive_endpoint_bracket)

Evaluate both endpoints exactly and test target-between-values in either orientation, including equality at an endpoint. A target outside the bracket does not satisfy this IVT premise; finite sample agreement does not establish continuity of an arbitrary function. The exact result, retaining the requested order, is ([-3, 1], 2, 4, -1, 1, 0, Yes, Yes, polynomial_continuity_plus_inclusive_endpoint_bracket).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r02-intermediate-value-theorem-0016/)

Intermediate value theorem · qf-mod47-ivt-hypothesis · candidate





## 17. Polynomial Continuity and Inclusive Endpoint Bracketing Are Separate Obligations

Coefficients=[-4, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=3<b=5 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Give (coefficients, a, b, value_at_a, value_at_b, target, polynomial_continuous, inclusive_bracket, certificate).

View answer and explanation([-4, 1], 3, 5, -1, 1, 0, Yes, Yes, polynomial_continuity_plus_inclusive_endpoint_bracket)

Evaluate both endpoints exactly and test target-between-values in either orientation, including equality at an endpoint. A target outside the bracket does not satisfy this IVT premise; finite sample agreement does not establish continuity of an arbitrary function. The exact result, retaining the requested order, is ([-4, 1], 3, 5, -1, 1, 0, Yes, Yes, polynomial_continuity_plus_inclusive_endpoint_bracket).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r02-intermediate-value-theorem-0017/)

Intermediate value theorem · qf-mod47-ivt-hypothesis · candidate





## 18. Polynomial Continuity and Inclusive Endpoint Bracketing Are Separate Obligations

Coefficients=[-5, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=4<b=6 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Give (coefficients, a, b, value_at_a, value_at_b, target, polynomial_continuous, inclusive_bracket, certificate).

View answer and explanation([-5, 1], 4, 6, -1, 1, 0, Yes, Yes, polynomial_continuity_plus_inclusive_endpoint_bracket)

Evaluate both endpoints exactly and test target-between-values in either orientation, including equality at an endpoint. A target outside the bracket does not satisfy this IVT premise; finite sample agreement does not establish continuity of an arbitrary function. The exact result, retaining the requested order, is ([-5, 1], 4, 6, -1, 1, 0, Yes, Yes, polynomial_continuity_plus_inclusive_endpoint_bracket).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r02-intermediate-value-theorem-0018/)

Intermediate value theorem · qf-mod47-ivt-hypothesis · candidate





## 19. Polynomial Continuity and Inclusive Endpoint Bracketing Are Separate Obligations

Coefficients=[-6, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=5<b=7 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Give (coefficients, a, b, value_at_a, value_at_b, target, polynomial_continuous, inclusive_bracket, certificate).

View answer and explanation([-6, 1], 5, 7, -1, 1, 0, Yes, Yes, polynomial_continuity_plus_inclusive_endpoint_bracket)

Evaluate both endpoints exactly and test target-between-values in either orientation, including equality at an endpoint. A target outside the bracket does not satisfy this IVT premise; finite sample agreement does not establish continuity of an arbitrary function. The exact result, retaining the requested order, is ([-6, 1], 5, 7, -1, 1, 0, Yes, Yes, polynomial_continuity_plus_inclusive_endpoint_bracket).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r02-intermediate-value-theorem-0019/)

Intermediate value theorem · qf-mod47-ivt-hypothesis · candidate





## 20. Polynomial Continuity and Inclusive Endpoint Bracketing Are Separate Obligations

Coefficients=[-7, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=6<b=8 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Give (coefficients, a, b, value_at_a, value_at_b, target, polynomial_continuous, inclusive_bracket, certificate).

View answer and explanation([-7, 1], 6, 8, -1, 1, 0, Yes, Yes, polynomial_continuity_plus_inclusive_endpoint_bracket)

Evaluate both endpoints exactly and test target-between-values in either orientation, including equality at an endpoint. A target outside the bracket does not satisfy this IVT premise; finite sample agreement does not establish continuity of an arbitrary function. The exact result, retaining the requested order, is ([-7, 1], 6, 8, -1, 1, 0, Yes, Yes, polynomial_continuity_plus_inclusive_endpoint_bracket).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r02-intermediate-value-theorem-0020/)

Intermediate value theorem · qf-mod47-ivt-hypothesis · candidate





## 21. Polynomial Continuity and Inclusive Endpoint Bracketing Are Separate Obligations

Coefficients=[-8, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=7<b=9 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Give (coefficients, a, b, value_at_a, value_at_b, target, polynomial_continuous, inclusive_bracket, certificate).

View answer and explanation([-8, 1], 7, 9, -1, 1, 0, Yes, Yes, polynomial_continuity_plus_inclusive_endpoint_bracket)

Evaluate both endpoints exactly and test target-between-values in either orientation, including equality at an endpoint. A target outside the bracket does not satisfy this IVT premise; finite sample agreement does not establish continuity of an arbitrary function. The exact result, retaining the requested order, is ([-8, 1], 7, 9, -1, 1, 0, Yes, Yes, polynomial_continuity_plus_inclusive_endpoint_bracket).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r02-intermediate-value-theorem-0021/)

Intermediate value theorem · qf-mod47-ivt-hypothesis · candidate





## 22. Polynomial Continuity and Inclusive Endpoint Bracketing Are Separate Obligations

Coefficients=[-9, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=8<b=10 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Give (coefficients, a, b, value_at_a, value_at_b, target, polynomial_continuous, inclusive_bracket, certificate).

View answer and explanation([-9, 1], 8, 10, -1, 1, 0, Yes, Yes, polynomial_continuity_plus_inclusive_endpoint_bracket)

Evaluate both endpoints exactly and test target-between-values in either orientation, including equality at an endpoint. A target outside the bracket does not satisfy this IVT premise; finite sample agreement does not establish continuity of an arbitrary function. The exact result, retaining the requested order, is ([-9, 1], 8, 10, -1, 1, 0, Yes, Yes, polynomial_continuity_plus_inclusive_endpoint_bracket).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r02-intermediate-value-theorem-0022/)

Intermediate value theorem · qf-mod47-ivt-hypothesis · candidate





## 23. Polynomial Continuity and Inclusive Endpoint Bracketing Are Separate Obligations

Coefficients=[-10, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=9<b=11 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Give (coefficients, a, b, value_at_a, value_at_b, target, polynomial_continuous, inclusive_bracket, certificate).

View answer and explanation([-10, 1], 9, 11, -1, 1, 0, Yes, Yes, polynomial_continuity_plus_inclusive_endpoint_bracket)

Evaluate both endpoints exactly and test target-between-values in either orientation, including equality at an endpoint. A target outside the bracket does not satisfy this IVT premise; finite sample agreement does not establish continuity of an arbitrary function. The exact result, retaining the requested order, is ([-10, 1], 9, 11, -1, 1, 0, Yes, Yes, polynomial_continuity_plus_inclusive_endpoint_bracket).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r02-intermediate-value-theorem-0023/)

Intermediate value theorem · qf-mod47-ivt-hypothesis · candidate





## 24. Polynomial Continuity and Inclusive Endpoint Bracketing Are Separate Obligations

Coefficients=[-11, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=10<b=12 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Give (coefficients, a, b, value_at_a, value_at_b, target, polynomial_continuous, inclusive_bracket, certificate).

View answer and explanation([-11, 1], 10, 12, -1, 1, 0, Yes, Yes, polynomial_continuity_plus_inclusive_endpoint_bracket)

Evaluate both endpoints exactly and test target-between-values in either orientation, including equality at an endpoint. A target outside the bracket does not satisfy this IVT premise; finite sample agreement does not establish continuity of an arbitrary function. The exact result, retaining the requested order, is ([-11, 1], 10, 12, -1, 1, 0, Yes, Yes, polynomial_continuity_plus_inclusive_endpoint_bracket).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r02-intermediate-value-theorem-0024/)

Intermediate value theorem · qf-mod47-ivt-hypothesis · candidate





## 25. Polynomial Continuity and Inclusive Endpoint Bracketing Are Separate Obligations

Coefficients=[-12, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=11<b=13 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Give (coefficients, a, b, value_at_a, value_at_b, target, polynomial_continuous, inclusive_bracket, certificate).

View answer and explanation([-12, 1], 11, 13, -1, 1, 0, Yes, Yes, polynomial_continuity_plus_inclusive_endpoint_bracket)

Evaluate both endpoints exactly and test target-between-values in either orientation, including equality at an endpoint. A target outside the bracket does not satisfy this IVT premise; finite sample agreement does not establish continuity of an arbitrary function. The exact result, retaining the requested order, is ([-12, 1], 11, 13, -1, 1, 0, Yes, Yes, polynomial_continuity_plus_inclusive_endpoint_bracket).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r02-intermediate-value-theorem-0025/)

Intermediate value theorem · qf-mod47-ivt-hypothesis · candidate





## 26. An Exact Closed-Interval Root Witness Does Not Assert Uniqueness

Coefficients=[12, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=-13<b=-11 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Supplied exact root=-12. Give (root, value_at_root, inside_closed_interval, hits_target, uniqueness_or_closed_form_claim, certificate).

View answer and explanation(-12, 0, Yes, Yes, No, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form)

Check membership in [a,b] and f(root)=target exactly, along with the bracket. The fixed No withholds uniqueness or a rational closed-form guarantee in the general theorem; it does not negate this supplied rational witness. The exact result, retaining the requested order, is (-12, 0, Yes, Yes, No, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r02-intermediate-value-theorem-0026/)

Intermediate value theorem · qf-mod47-ivt-exact-root · candidate





## 27. An Exact Closed-Interval Root Witness Does Not Assert Uniqueness

Coefficients=[11, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=-12<b=-10 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Supplied exact root=-11. Give (root, value_at_root, inside_closed_interval, hits_target, uniqueness_or_closed_form_claim, certificate).

View answer and explanation(-11, 0, Yes, Yes, No, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form)

Check membership in [a,b] and f(root)=target exactly, along with the bracket. The fixed No withholds uniqueness or a rational closed-form guarantee in the general theorem; it does not negate this supplied rational witness. The exact result, retaining the requested order, is (-11, 0, Yes, Yes, No, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r02-intermediate-value-theorem-0027/)

Intermediate value theorem · qf-mod47-ivt-exact-root · candidate





## 28. An Exact Closed-Interval Root Witness Does Not Assert Uniqueness

Coefficients=[10, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=-11<b=-9 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Supplied exact root=-10. Give (root, value_at_root, inside_closed_interval, hits_target, uniqueness_or_closed_form_claim, certificate).

View answer and explanation(-10, 0, Yes, Yes, No, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form)

Check membership in [a,b] and f(root)=target exactly, along with the bracket. The fixed No withholds uniqueness or a rational closed-form guarantee in the general theorem; it does not negate this supplied rational witness. The exact result, retaining the requested order, is (-10, 0, Yes, Yes, No, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r02-intermediate-value-theorem-0028/)

Intermediate value theorem · qf-mod47-ivt-exact-root · candidate





## 29. An Exact Closed-Interval Root Witness Does Not Assert Uniqueness

Coefficients=[9, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=-10<b=-8 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Supplied exact root=-9. Give (root, value_at_root, inside_closed_interval, hits_target, uniqueness_or_closed_form_claim, certificate).

View answer and explanation(-9, 0, Yes, Yes, No, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form)

Check membership in [a,b] and f(root)=target exactly, along with the bracket. The fixed No withholds uniqueness or a rational closed-form guarantee in the general theorem; it does not negate this supplied rational witness. The exact result, retaining the requested order, is (-9, 0, Yes, Yes, No, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r02-intermediate-value-theorem-0029/)

Intermediate value theorem · qf-mod47-ivt-exact-root · candidate





## 30. An Exact Closed-Interval Root Witness Does Not Assert Uniqueness

Coefficients=[8, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=-9<b=-7 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Supplied exact root=-8. Give (root, value_at_root, inside_closed_interval, hits_target, uniqueness_or_closed_form_claim, certificate).

View answer and explanation(-8, 0, Yes, Yes, No, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form)

Check membership in [a,b] and f(root)=target exactly, along with the bracket. The fixed No withholds uniqueness or a rational closed-form guarantee in the general theorem; it does not negate this supplied rational witness. The exact result, retaining the requested order, is (-8, 0, Yes, Yes, No, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r02-intermediate-value-theorem-0030/)

Intermediate value theorem · qf-mod47-ivt-exact-root · candidate





## 31. An Exact Closed-Interval Root Witness Does Not Assert Uniqueness

Coefficients=[7, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=-8<b=-6 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Supplied exact root=-7. Give (root, value_at_root, inside_closed_interval, hits_target, uniqueness_or_closed_form_claim, certificate).

View answer and explanation(-7, 0, Yes, Yes, No, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form)

Check membership in [a,b] and f(root)=target exactly, along with the bracket. The fixed No withholds uniqueness or a rational closed-form guarantee in the general theorem; it does not negate this supplied rational witness. The exact result, retaining the requested order, is (-7, 0, Yes, Yes, No, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r02-intermediate-value-theorem-0031/)

Intermediate value theorem · qf-mod47-ivt-exact-root · candidate





## 32. An Exact Closed-Interval Root Witness Does Not Assert Uniqueness

Coefficients=[6, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=-7<b=-5 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Supplied exact root=-6. Give (root, value_at_root, inside_closed_interval, hits_target, uniqueness_or_closed_form_claim, certificate).

View answer and explanation(-6, 0, Yes, Yes, No, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form)

Check membership in [a,b] and f(root)=target exactly, along with the bracket. The fixed No withholds uniqueness or a rational closed-form guarantee in the general theorem; it does not negate this supplied rational witness. The exact result, retaining the requested order, is (-6, 0, Yes, Yes, No, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r02-intermediate-value-theorem-0032/)

Intermediate value theorem · qf-mod47-ivt-exact-root · candidate





## 33. An Exact Closed-Interval Root Witness Does Not Assert Uniqueness

Coefficients=[5, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=-6<b=-4 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Supplied exact root=-5. Give (root, value_at_root, inside_closed_interval, hits_target, uniqueness_or_closed_form_claim, certificate).

View answer and explanation(-5, 0, Yes, Yes, No, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form)

Check membership in [a,b] and f(root)=target exactly, along with the bracket. The fixed No withholds uniqueness or a rational closed-form guarantee in the general theorem; it does not negate this supplied rational witness. The exact result, retaining the requested order, is (-5, 0, Yes, Yes, No, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r02-intermediate-value-theorem-0033/)

Intermediate value theorem · qf-mod47-ivt-exact-root · candidate





## 34. An Exact Closed-Interval Root Witness Does Not Assert Uniqueness

Coefficients=[4, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=-5<b=-3 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Supplied exact root=-4. Give (root, value_at_root, inside_closed_interval, hits_target, uniqueness_or_closed_form_claim, certificate).

View answer and explanation(-4, 0, Yes, Yes, No, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form)

Check membership in [a,b] and f(root)=target exactly, along with the bracket. The fixed No withholds uniqueness or a rational closed-form guarantee in the general theorem; it does not negate this supplied rational witness. The exact result, retaining the requested order, is (-4, 0, Yes, Yes, No, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r02-intermediate-value-theorem-0034/)

Intermediate value theorem · qf-mod47-ivt-exact-root · candidate





## 35. An Exact Closed-Interval Root Witness Does Not Assert Uniqueness

Coefficients=[3, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=-4<b=-2 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Supplied exact root=-3. Give (root, value_at_root, inside_closed_interval, hits_target, uniqueness_or_closed_form_claim, certificate).

View answer and explanation(-3, 0, Yes, Yes, No, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form)

Check membership in [a,b] and f(root)=target exactly, along with the bracket. The fixed No withholds uniqueness or a rational closed-form guarantee in the general theorem; it does not negate this supplied rational witness. The exact result, retaining the requested order, is (-3, 0, Yes, Yes, No, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r02-intermediate-value-theorem-0035/)

Intermediate value theorem · qf-mod47-ivt-exact-root · candidate





## 36. An Exact Closed-Interval Root Witness Does Not Assert Uniqueness

Coefficients=[2, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=-3<b=-1 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Supplied exact root=-2. Give (root, value_at_root, inside_closed_interval, hits_target, uniqueness_or_closed_form_claim, certificate).

View answer and explanation(-2, 0, Yes, Yes, No, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form)

Check membership in [a,b] and f(root)=target exactly, along with the bracket. The fixed No withholds uniqueness or a rational closed-form guarantee in the general theorem; it does not negate this supplied rational witness. The exact result, retaining the requested order, is (-2, 0, Yes, Yes, No, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r02-intermediate-value-theorem-0036/)

Intermediate value theorem · qf-mod47-ivt-exact-root · candidate





## 37. An Exact Closed-Interval Root Witness Does Not Assert Uniqueness

Coefficients=[1, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=-2<b=0 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Supplied exact root=-1. Give (root, value_at_root, inside_closed_interval, hits_target, uniqueness_or_closed_form_claim, certificate).

View answer and explanation(-1, 0, Yes, Yes, No, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form)

Check membership in [a,b] and f(root)=target exactly, along with the bracket. The fixed No withholds uniqueness or a rational closed-form guarantee in the general theorem; it does not negate this supplied rational witness. The exact result, retaining the requested order, is (-1, 0, Yes, Yes, No, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r02-intermediate-value-theorem-0037/)

Intermediate value theorem · qf-mod47-ivt-exact-root · candidate





## 38. An Exact Closed-Interval Root Witness Does Not Assert Uniqueness

Coefficients=[0, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=-1<b=1 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Supplied exact root=0. Give (root, value_at_root, inside_closed_interval, hits_target, uniqueness_or_closed_form_claim, certificate).

View answer and explanation(0, 0, Yes, Yes, No, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form)

Check membership in [a,b] and f(root)=target exactly, along with the bracket. The fixed No withholds uniqueness or a rational closed-form guarantee in the general theorem; it does not negate this supplied rational witness. The exact result, retaining the requested order, is (0, 0, Yes, Yes, No, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r02-intermediate-value-theorem-0038/)

Intermediate value theorem · qf-mod47-ivt-exact-root · candidate





## 39. An Exact Closed-Interval Root Witness Does Not Assert Uniqueness

Coefficients=[-1, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=0<b=2 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Supplied exact root=1. Give (root, value_at_root, inside_closed_interval, hits_target, uniqueness_or_closed_form_claim, certificate).

View answer and explanation(1, 0, Yes, Yes, No, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form)

Check membership in [a,b] and f(root)=target exactly, along with the bracket. The fixed No withholds uniqueness or a rational closed-form guarantee in the general theorem; it does not negate this supplied rational witness. The exact result, retaining the requested order, is (1, 0, Yes, Yes, No, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r02-intermediate-value-theorem-0039/)

Intermediate value theorem · qf-mod47-ivt-exact-root · candidate





## 40. An Exact Closed-Interval Root Witness Does Not Assert Uniqueness

Coefficients=[-2, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=1<b=3 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Supplied exact root=2. Give (root, value_at_root, inside_closed_interval, hits_target, uniqueness_or_closed_form_claim, certificate).

View answer and explanation(2, 0, Yes, Yes, No, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form)

Check membership in [a,b] and f(root)=target exactly, along with the bracket. The fixed No withholds uniqueness or a rational closed-form guarantee in the general theorem; it does not negate this supplied rational witness. The exact result, retaining the requested order, is (2, 0, Yes, Yes, No, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r02-intermediate-value-theorem-0040/)

Intermediate value theorem · qf-mod47-ivt-exact-root · candidate





## 41. An Exact Closed-Interval Root Witness Does Not Assert Uniqueness

Coefficients=[-3, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=2<b=4 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Supplied exact root=3. Give (root, value_at_root, inside_closed_interval, hits_target, uniqueness_or_closed_form_claim, certificate).

View answer and explanation(3, 0, Yes, Yes, No, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form)

Check membership in [a,b] and f(root)=target exactly, along with the bracket. The fixed No withholds uniqueness or a rational closed-form guarantee in the general theorem; it does not negate this supplied rational witness. The exact result, retaining the requested order, is (3, 0, Yes, Yes, No, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r02-intermediate-value-theorem-0041/)

Intermediate value theorem · qf-mod47-ivt-exact-root · candidate





## 42. An Exact Closed-Interval Root Witness Does Not Assert Uniqueness

Coefficients=[-4, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=3<b=5 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Supplied exact root=4. Give (root, value_at_root, inside_closed_interval, hits_target, uniqueness_or_closed_form_claim, certificate).

View answer and explanation(4, 0, Yes, Yes, No, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form)

Check membership in [a,b] and f(root)=target exactly, along with the bracket. The fixed No withholds uniqueness or a rational closed-form guarantee in the general theorem; it does not negate this supplied rational witness. The exact result, retaining the requested order, is (4, 0, Yes, Yes, No, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r02-intermediate-value-theorem-0042/)

Intermediate value theorem · qf-mod47-ivt-exact-root · candidate





## 43. An Exact Closed-Interval Root Witness Does Not Assert Uniqueness

Coefficients=[-5, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=4<b=6 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Supplied exact root=5. Give (root, value_at_root, inside_closed_interval, hits_target, uniqueness_or_closed_form_claim, certificate).

View answer and explanation(5, 0, Yes, Yes, No, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form)

Check membership in [a,b] and f(root)=target exactly, along with the bracket. The fixed No withholds uniqueness or a rational closed-form guarantee in the general theorem; it does not negate this supplied rational witness. The exact result, retaining the requested order, is (5, 0, Yes, Yes, No, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r02-intermediate-value-theorem-0043/)

Intermediate value theorem · qf-mod47-ivt-exact-root · candidate





## 44. An Exact Closed-Interval Root Witness Does Not Assert Uniqueness

Coefficients=[-6, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=5<b=7 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Supplied exact root=6. Give (root, value_at_root, inside_closed_interval, hits_target, uniqueness_or_closed_form_claim, certificate).

View answer and explanation(6, 0, Yes, Yes, No, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form)

Check membership in [a,b] and f(root)=target exactly, along with the bracket. The fixed No withholds uniqueness or a rational closed-form guarantee in the general theorem; it does not negate this supplied rational witness. The exact result, retaining the requested order, is (6, 0, Yes, Yes, No, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r02-intermediate-value-theorem-0044/)

Intermediate value theorem · qf-mod47-ivt-exact-root · candidate





## 45. An Exact Closed-Interval Root Witness Does Not Assert Uniqueness

Coefficients=[-7, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=6<b=8 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Supplied exact root=7. Give (root, value_at_root, inside_closed_interval, hits_target, uniqueness_or_closed_form_claim, certificate).

View answer and explanation(7, 0, Yes, Yes, No, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form)

Check membership in [a,b] and f(root)=target exactly, along with the bracket. The fixed No withholds uniqueness or a rational closed-form guarantee in the general theorem; it does not negate this supplied rational witness. The exact result, retaining the requested order, is (7, 0, Yes, Yes, No, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r02-intermediate-value-theorem-0045/)

Intermediate value theorem · qf-mod47-ivt-exact-root · candidate





## 46. An Exact Closed-Interval Root Witness Does Not Assert Uniqueness

Coefficients=[-8, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=7<b=9 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Supplied exact root=8. Give (root, value_at_root, inside_closed_interval, hits_target, uniqueness_or_closed_form_claim, certificate).

View answer and explanation(8, 0, Yes, Yes, No, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form)

Check membership in [a,b] and f(root)=target exactly, along with the bracket. The fixed No withholds uniqueness or a rational closed-form guarantee in the general theorem; it does not negate this supplied rational witness. The exact result, retaining the requested order, is (8, 0, Yes, Yes, No, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r02-intermediate-value-theorem-0046/)

Intermediate value theorem · qf-mod47-ivt-exact-root · candidate





## 47. An Exact Closed-Interval Root Witness Does Not Assert Uniqueness

Coefficients=[-9, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=8<b=10 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Supplied exact root=9. Give (root, value_at_root, inside_closed_interval, hits_target, uniqueness_or_closed_form_claim, certificate).

View answer and explanation(9, 0, Yes, Yes, No, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form)

Check membership in [a,b] and f(root)=target exactly, along with the bracket. The fixed No withholds uniqueness or a rational closed-form guarantee in the general theorem; it does not negate this supplied rational witness. The exact result, retaining the requested order, is (9, 0, Yes, Yes, No, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r02-intermediate-value-theorem-0047/)

Intermediate value theorem · qf-mod47-ivt-exact-root · candidate





## 48. An Exact Closed-Interval Root Witness Does Not Assert Uniqueness

Coefficients=[-10, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=9<b=11 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Supplied exact root=10. Give (root, value_at_root, inside_closed_interval, hits_target, uniqueness_or_closed_form_claim, certificate).

View answer and explanation(10, 0, Yes, Yes, No, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form)

Check membership in [a,b] and f(root)=target exactly, along with the bracket. The fixed No withholds uniqueness or a rational closed-form guarantee in the general theorem; it does not negate this supplied rational witness. The exact result, retaining the requested order, is (10, 0, Yes, Yes, No, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r02-intermediate-value-theorem-0048/)

Intermediate value theorem · qf-mod47-ivt-exact-root · candidate





## 49. An Exact Closed-Interval Root Witness Does Not Assert Uniqueness

Coefficients=[-11, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=10<b=12 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Supplied exact root=11. Give (root, value_at_root, inside_closed_interval, hits_target, uniqueness_or_closed_form_claim, certificate).

View answer and explanation(11, 0, Yes, Yes, No, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form)

Check membership in [a,b] and f(root)=target exactly, along with the bracket. The fixed No withholds uniqueness or a rational closed-form guarantee in the general theorem; it does not negate this supplied rational witness. The exact result, retaining the requested order, is (11, 0, Yes, Yes, No, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r02-intermediate-value-theorem-0049/)

Intermediate value theorem · qf-mod47-ivt-exact-root · candidate





## 50. An Exact Closed-Interval Root Witness Does Not Assert Uniqueness

Coefficients=[-12, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=11<b=13 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Supplied exact root=12. Give (root, value_at_root, inside_closed_interval, hits_target, uniqueness_or_closed_form_claim, certificate).

View answer and explanation(12, 0, Yes, Yes, No, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form)

Check membership in [a,b] and f(root)=target exactly, along with the bracket. The fixed No withholds uniqueness or a rational closed-form guarantee in the general theorem; it does not negate this supplied rational witness. The exact result, retaining the requested order, is (12, 0, Yes, Yes, No, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r02-intermediate-value-theorem-0050/)

Intermediate value theorem · qf-mod47-ivt-exact-root · candidate





## 51. The Inclusive Bracket Allows an Endpoint Root

Coefficients=[13, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=-13<b=-11 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Supplied exact root=-13. Give (a, b, target, root, root_is_endpoint, inclusive_bracket, root_certificate, certificate).

View answer and explanation(-13, -11, 0, -13, Yes, Yes, Yes, endpoints_are_included)

Retain endpoint/root equality as a separate flag. The closed interval includes either endpoint, but an interior root can also satisfy the same family. An exact witness is not a claim that finite bisection has converged. The exact result, retaining the requested order, is (-13, -11, 0, -13, Yes, Yes, Yes, endpoints_are_included).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r02-intermediate-value-theorem-0051/)

Intermediate value theorem · qf-mod47-ivt-endpoint-bracket · candidate





## 52. The Inclusive Bracket Allows an Endpoint Root

Coefficients=[12, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=-12<b=-10 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Supplied exact root=-12. Give (a, b, target, root, root_is_endpoint, inclusive_bracket, root_certificate, certificate).

View answer and explanation(-12, -10, 0, -12, Yes, Yes, Yes, endpoints_are_included)

Retain endpoint/root equality as a separate flag. The closed interval includes either endpoint, but an interior root can also satisfy the same family. An exact witness is not a claim that finite bisection has converged. The exact result, retaining the requested order, is (-12, -10, 0, -12, Yes, Yes, Yes, endpoints_are_included).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r02-intermediate-value-theorem-0052/)

Intermediate value theorem · qf-mod47-ivt-endpoint-bracket · candidate





## 53. The Inclusive Bracket Allows an Endpoint Root

Coefficients=[11, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=-11<b=-9 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Supplied exact root=-11. Give (a, b, target, root, root_is_endpoint, inclusive_bracket, root_certificate, certificate).

View answer and explanation(-11, -9, 0, -11, Yes, Yes, Yes, endpoints_are_included)

Retain endpoint/root equality as a separate flag. The closed interval includes either endpoint, but an interior root can also satisfy the same family. An exact witness is not a claim that finite bisection has converged. The exact result, retaining the requested order, is (-11, -9, 0, -11, Yes, Yes, Yes, endpoints_are_included).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r02-intermediate-value-theorem-0053/)

Intermediate value theorem · qf-mod47-ivt-endpoint-bracket · candidate





## 54. The Inclusive Bracket Allows an Endpoint Root

Coefficients=[10, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=-10<b=-8 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Supplied exact root=-10. Give (a, b, target, root, root_is_endpoint, inclusive_bracket, root_certificate, certificate).

View answer and explanation(-10, -8, 0, -10, Yes, Yes, Yes, endpoints_are_included)

Retain endpoint/root equality as a separate flag. The closed interval includes either endpoint, but an interior root can also satisfy the same family. An exact witness is not a claim that finite bisection has converged. The exact result, retaining the requested order, is (-10, -8, 0, -10, Yes, Yes, Yes, endpoints_are_included).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r02-intermediate-value-theorem-0054/)

Intermediate value theorem · qf-mod47-ivt-endpoint-bracket · candidate





## 55. The Inclusive Bracket Allows an Endpoint Root

Coefficients=[9, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=-9<b=-7 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Supplied exact root=-9. Give (a, b, target, root, root_is_endpoint, inclusive_bracket, root_certificate, certificate).

View answer and explanation(-9, -7, 0, -9, Yes, Yes, Yes, endpoints_are_included)

Retain endpoint/root equality as a separate flag. The closed interval includes either endpoint, but an interior root can also satisfy the same family. An exact witness is not a claim that finite bisection has converged. The exact result, retaining the requested order, is (-9, -7, 0, -9, Yes, Yes, Yes, endpoints_are_included).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r02-intermediate-value-theorem-0055/)

Intermediate value theorem · qf-mod47-ivt-endpoint-bracket · candidate





## 56. The Inclusive Bracket Allows an Endpoint Root

Coefficients=[8, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=-8<b=-6 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Supplied exact root=-8. Give (a, b, target, root, root_is_endpoint, inclusive_bracket, root_certificate, certificate).

View answer and explanation(-8, -6, 0, -8, Yes, Yes, Yes, endpoints_are_included)

Retain endpoint/root equality as a separate flag. The closed interval includes either endpoint, but an interior root can also satisfy the same family. An exact witness is not a claim that finite bisection has converged. The exact result, retaining the requested order, is (-8, -6, 0, -8, Yes, Yes, Yes, endpoints_are_included).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r02-intermediate-value-theorem-0056/)

Intermediate value theorem · qf-mod47-ivt-endpoint-bracket · candidate





## 57. The Inclusive Bracket Allows an Endpoint Root

Coefficients=[7, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=-7<b=-5 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Supplied exact root=-7. Give (a, b, target, root, root_is_endpoint, inclusive_bracket, root_certificate, certificate).

View answer and explanation(-7, -5, 0, -7, Yes, Yes, Yes, endpoints_are_included)

Retain endpoint/root equality as a separate flag. The closed interval includes either endpoint, but an interior root can also satisfy the same family. An exact witness is not a claim that finite bisection has converged. The exact result, retaining the requested order, is (-7, -5, 0, -7, Yes, Yes, Yes, endpoints_are_included).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r02-intermediate-value-theorem-0057/)

Intermediate value theorem · qf-mod47-ivt-endpoint-bracket · candidate





## 58. The Inclusive Bracket Allows an Endpoint Root

Coefficients=[6, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=-6<b=-4 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Supplied exact root=-6. Give (a, b, target, root, root_is_endpoint, inclusive_bracket, root_certificate, certificate).

View answer and explanation(-6, -4, 0, -6, Yes, Yes, Yes, endpoints_are_included)

Retain endpoint/root equality as a separate flag. The closed interval includes either endpoint, but an interior root can also satisfy the same family. An exact witness is not a claim that finite bisection has converged. The exact result, retaining the requested order, is (-6, -4, 0, -6, Yes, Yes, Yes, endpoints_are_included).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r02-intermediate-value-theorem-0058/)

Intermediate value theorem · qf-mod47-ivt-endpoint-bracket · candidate





## 59. The Inclusive Bracket Allows an Endpoint Root

Coefficients=[5, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=-5<b=-3 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Supplied exact root=-5. Give (a, b, target, root, root_is_endpoint, inclusive_bracket, root_certificate, certificate).

View answer and explanation(-5, -3, 0, -5, Yes, Yes, Yes, endpoints_are_included)

Retain endpoint/root equality as a separate flag. The closed interval includes either endpoint, but an interior root can also satisfy the same family. An exact witness is not a claim that finite bisection has converged. The exact result, retaining the requested order, is (-5, -3, 0, -5, Yes, Yes, Yes, endpoints_are_included).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r02-intermediate-value-theorem-0059/)

Intermediate value theorem · qf-mod47-ivt-endpoint-bracket · candidate





## 60. The Inclusive Bracket Allows an Endpoint Root

Coefficients=[4, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=-4<b=-2 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Supplied exact root=-4. Give (a, b, target, root, root_is_endpoint, inclusive_bracket, root_certificate, certificate).

View answer and explanation(-4, -2, 0, -4, Yes, Yes, Yes, endpoints_are_included)

Retain endpoint/root equality as a separate flag. The closed interval includes either endpoint, but an interior root can also satisfy the same family. An exact witness is not a claim that finite bisection has converged. The exact result, retaining the requested order, is (-4, -2, 0, -4, Yes, Yes, Yes, endpoints_are_included).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r02-intermediate-value-theorem-0060/)

Intermediate value theorem · qf-mod47-ivt-endpoint-bracket · candidate





## 61. The Inclusive Bracket Allows an Endpoint Root

Coefficients=[3, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=-3<b=-1 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Supplied exact root=-3. Give (a, b, target, root, root_is_endpoint, inclusive_bracket, root_certificate, certificate).

View answer and explanation(-3, -1, 0, -3, Yes, Yes, Yes, endpoints_are_included)

Retain endpoint/root equality as a separate flag. The closed interval includes either endpoint, but an interior root can also satisfy the same family. An exact witness is not a claim that finite bisection has converged. The exact result, retaining the requested order, is (-3, -1, 0, -3, Yes, Yes, Yes, endpoints_are_included).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r02-intermediate-value-theorem-0061/)

Intermediate value theorem · qf-mod47-ivt-endpoint-bracket · candidate





## 62. The Inclusive Bracket Allows an Endpoint Root

Coefficients=[2, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=-2<b=0 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Supplied exact root=-2. Give (a, b, target, root, root_is_endpoint, inclusive_bracket, root_certificate, certificate).

View answer and explanation(-2, 0, 0, -2, Yes, Yes, Yes, endpoints_are_included)

Retain endpoint/root equality as a separate flag. The closed interval includes either endpoint, but an interior root can also satisfy the same family. An exact witness is not a claim that finite bisection has converged. The exact result, retaining the requested order, is (-2, 0, 0, -2, Yes, Yes, Yes, endpoints_are_included).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r02-intermediate-value-theorem-0062/)

Intermediate value theorem · qf-mod47-ivt-endpoint-bracket · candidate





## 63. The Inclusive Bracket Allows an Endpoint Root

Coefficients=[1, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=-1<b=1 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Supplied exact root=-1. Give (a, b, target, root, root_is_endpoint, inclusive_bracket, root_certificate, certificate).

View answer and explanation(-1, 1, 0, -1, Yes, Yes, Yes, endpoints_are_included)

Retain endpoint/root equality as a separate flag. The closed interval includes either endpoint, but an interior root can also satisfy the same family. An exact witness is not a claim that finite bisection has converged. The exact result, retaining the requested order, is (-1, 1, 0, -1, Yes, Yes, Yes, endpoints_are_included).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r02-intermediate-value-theorem-0063/)

Intermediate value theorem · qf-mod47-ivt-endpoint-bracket · candidate





## 64. The Inclusive Bracket Allows an Endpoint Root

Coefficients=[0, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=0<b=2 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Supplied exact root=0. Give (a, b, target, root, root_is_endpoint, inclusive_bracket, root_certificate, certificate).

View answer and explanation(0, 2, 0, 0, Yes, Yes, Yes, endpoints_are_included)

Retain endpoint/root equality as a separate flag. The closed interval includes either endpoint, but an interior root can also satisfy the same family. An exact witness is not a claim that finite bisection has converged. The exact result, retaining the requested order, is (0, 2, 0, 0, Yes, Yes, Yes, endpoints_are_included).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r02-intermediate-value-theorem-0064/)

Intermediate value theorem · qf-mod47-ivt-endpoint-bracket · candidate





## 65. The Inclusive Bracket Allows an Endpoint Root

Coefficients=[-1, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=1<b=3 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Supplied exact root=1. Give (a, b, target, root, root_is_endpoint, inclusive_bracket, root_certificate, certificate).

View answer and explanation(1, 3, 0, 1, Yes, Yes, Yes, endpoints_are_included)

Retain endpoint/root equality as a separate flag. The closed interval includes either endpoint, but an interior root can also satisfy the same family. An exact witness is not a claim that finite bisection has converged. The exact result, retaining the requested order, is (1, 3, 0, 1, Yes, Yes, Yes, endpoints_are_included).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r02-intermediate-value-theorem-0065/)

Intermediate value theorem · qf-mod47-ivt-endpoint-bracket · candidate





## 66. The Inclusive Bracket Allows an Endpoint Root

Coefficients=[-2, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=2<b=4 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Supplied exact root=2. Give (a, b, target, root, root_is_endpoint, inclusive_bracket, root_certificate, certificate).

View answer and explanation(2, 4, 0, 2, Yes, Yes, Yes, endpoints_are_included)

Retain endpoint/root equality as a separate flag. The closed interval includes either endpoint, but an interior root can also satisfy the same family. An exact witness is not a claim that finite bisection has converged. The exact result, retaining the requested order, is (2, 4, 0, 2, Yes, Yes, Yes, endpoints_are_included).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r02-intermediate-value-theorem-0066/)

Intermediate value theorem · qf-mod47-ivt-endpoint-bracket · candidate





## 67. The Inclusive Bracket Allows an Endpoint Root

Coefficients=[-3, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=3<b=5 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Supplied exact root=3. Give (a, b, target, root, root_is_endpoint, inclusive_bracket, root_certificate, certificate).

View answer and explanation(3, 5, 0, 3, Yes, Yes, Yes, endpoints_are_included)

Retain endpoint/root equality as a separate flag. The closed interval includes either endpoint, but an interior root can also satisfy the same family. An exact witness is not a claim that finite bisection has converged. The exact result, retaining the requested order, is (3, 5, 0, 3, Yes, Yes, Yes, endpoints_are_included).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r02-intermediate-value-theorem-0067/)

Intermediate value theorem · qf-mod47-ivt-endpoint-bracket · candidate





## 68. The Inclusive Bracket Allows an Endpoint Root

Coefficients=[-4, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=4<b=6 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Supplied exact root=4. Give (a, b, target, root, root_is_endpoint, inclusive_bracket, root_certificate, certificate).

View answer and explanation(4, 6, 0, 4, Yes, Yes, Yes, endpoints_are_included)

Retain endpoint/root equality as a separate flag. The closed interval includes either endpoint, but an interior root can also satisfy the same family. An exact witness is not a claim that finite bisection has converged. The exact result, retaining the requested order, is (4, 6, 0, 4, Yes, Yes, Yes, endpoints_are_included).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r02-intermediate-value-theorem-0068/)

Intermediate value theorem · qf-mod47-ivt-endpoint-bracket · candidate





## 69. The Inclusive Bracket Allows an Endpoint Root

Coefficients=[-5, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=5<b=7 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Supplied exact root=5. Give (a, b, target, root, root_is_endpoint, inclusive_bracket, root_certificate, certificate).

View answer and explanation(5, 7, 0, 5, Yes, Yes, Yes, endpoints_are_included)

Retain endpoint/root equality as a separate flag. The closed interval includes either endpoint, but an interior root can also satisfy the same family. An exact witness is not a claim that finite bisection has converged. The exact result, retaining the requested order, is (5, 7, 0, 5, Yes, Yes, Yes, endpoints_are_included).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r02-intermediate-value-theorem-0069/)

Intermediate value theorem · qf-mod47-ivt-endpoint-bracket · candidate





## 70. The Inclusive Bracket Allows an Endpoint Root

Coefficients=[-6, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=6<b=8 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Supplied exact root=6. Give (a, b, target, root, root_is_endpoint, inclusive_bracket, root_certificate, certificate).

View answer and explanation(6, 8, 0, 6, Yes, Yes, Yes, endpoints_are_included)

Retain endpoint/root equality as a separate flag. The closed interval includes either endpoint, but an interior root can also satisfy the same family. An exact witness is not a claim that finite bisection has converged. The exact result, retaining the requested order, is (6, 8, 0, 6, Yes, Yes, Yes, endpoints_are_included).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r02-intermediate-value-theorem-0070/)

Intermediate value theorem · qf-mod47-ivt-endpoint-bracket · candidate





## 71. The Inclusive Bracket Allows an Endpoint Root

Coefficients=[-7, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=7<b=9 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Supplied exact root=7. Give (a, b, target, root, root_is_endpoint, inclusive_bracket, root_certificate, certificate).

View answer and explanation(7, 9, 0, 7, Yes, Yes, Yes, endpoints_are_included)

Retain endpoint/root equality as a separate flag. The closed interval includes either endpoint, but an interior root can also satisfy the same family. An exact witness is not a claim that finite bisection has converged. The exact result, retaining the requested order, is (7, 9, 0, 7, Yes, Yes, Yes, endpoints_are_included).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r02-intermediate-value-theorem-0071/)

Intermediate value theorem · qf-mod47-ivt-endpoint-bracket · candidate





## 72. The Inclusive Bracket Allows an Endpoint Root

Coefficients=[-8, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=8<b=10 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Supplied exact root=8. Give (a, b, target, root, root_is_endpoint, inclusive_bracket, root_certificate, certificate).

View answer and explanation(8, 10, 0, 8, Yes, Yes, Yes, endpoints_are_included)

Retain endpoint/root equality as a separate flag. The closed interval includes either endpoint, but an interior root can also satisfy the same family. An exact witness is not a claim that finite bisection has converged. The exact result, retaining the requested order, is (8, 10, 0, 8, Yes, Yes, Yes, endpoints_are_included).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r02-intermediate-value-theorem-0072/)

Intermediate value theorem · qf-mod47-ivt-endpoint-bracket · candidate





## 73. The Inclusive Bracket Allows an Endpoint Root

Coefficients=[-9, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=9<b=11 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Supplied exact root=9. Give (a, b, target, root, root_is_endpoint, inclusive_bracket, root_certificate, certificate).

View answer and explanation(9, 11, 0, 9, Yes, Yes, Yes, endpoints_are_included)

Retain endpoint/root equality as a separate flag. The closed interval includes either endpoint, but an interior root can also satisfy the same family. An exact witness is not a claim that finite bisection has converged. The exact result, retaining the requested order, is (9, 11, 0, 9, Yes, Yes, Yes, endpoints_are_included).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r02-intermediate-value-theorem-0073/)

Intermediate value theorem · qf-mod47-ivt-endpoint-bracket · candidate





## 74. The Inclusive Bracket Allows an Endpoint Root

Coefficients=[-10, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=10<b=12 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Supplied exact root=10. Give (a, b, target, root, root_is_endpoint, inclusive_bracket, root_certificate, certificate).

View answer and explanation(10, 12, 0, 10, Yes, Yes, Yes, endpoints_are_included)

Retain endpoint/root equality as a separate flag. The closed interval includes either endpoint, but an interior root can also satisfy the same family. An exact witness is not a claim that finite bisection has converged. The exact result, retaining the requested order, is (10, 12, 0, 10, Yes, Yes, Yes, endpoints_are_included).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r02-intermediate-value-theorem-0074/)

Intermediate value theorem · qf-mod47-ivt-endpoint-bracket · candidate





## 75. The Inclusive Bracket Allows an Endpoint Root

Coefficients=[-11, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=11<b=13 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Supplied exact root=11. Give (a, b, target, root, root_is_endpoint, inclusive_bracket, root_certificate, certificate).

View answer and explanation(11, 13, 0, 11, Yes, Yes, Yes, endpoints_are_included)

Retain endpoint/root equality as a separate flag. The closed interval includes either endpoint, but an interior root can also satisfy the same family. An exact witness is not a claim that finite bisection has converged. The exact result, retaining the requested order, is (11, 13, 0, 11, Yes, Yes, Yes, endpoints_are_included).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r02-intermediate-value-theorem-0075/)

Intermediate value theorem · qf-mod47-ivt-endpoint-bracket · candidate





## 76. A Discontinuous Sign Jump and Failure of Uniqueness Have Different Boundaries

Use boundary mode=discontinuous_jump and positive exact scale=1. Give (mode, function_description, a, b, target, inclusive_bracket, continuous, root_exists).

The three modes have8/7/8 fields: jump has no root, nonbracket fails the bracket, and nonunique has two endpoint roots. The final No has different roles; arity cannot choose the branch.

View answer and explanation(discontinuous_jump, f(x)=-1_for_x<0_and_1_for_x>=0, -1, 1, 0, Yes, No, No)

The discontinuous jump takes only ±scale and misses0 despite bracketing it on [−1,1]. A scaled identity on [0,1] misses target2*scale, so the bracket fails. For x^2 on [−scale,scale], target scale^2 has both endpoint roots: existence is not uniqueness. Preserve each branch field meaning and do not infer continuity from samples. The exact result, retaining the requested order, is (discontinuous_jump, f(x)=-1_for_x<0_and_1_for_x>=0, -1, 1, 0, Yes, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r02-intermediate-value-theorem-0076/)

Intermediate value theorem · qf-mod47-ivt-discontinuity-boundary · candidate





## 77. A Discontinuous Sign Jump and Failure of Uniqueness Have Different Boundaries

Use boundary mode=nonbracket and positive exact scale=2. Give (mode, function_description, a, b, target, inclusive_bracket, root_exists).

The three modes have8/7/8 fields: jump has no root, nonbracket fails the bracket, and nonunique has two endpoint roots. The final No has different roles; arity cannot choose the branch.

View answer and explanation(nonbracket, f(x)=2*x, 0, 1, 4, No, No)

The discontinuous jump takes only ±scale and misses0 despite bracketing it on [−1,1]. A scaled identity on [0,1] misses target2*scale, so the bracket fails. For x^2 on [−scale,scale], target scale^2 has both endpoint roots: existence is not uniqueness. Preserve each branch field meaning and do not infer continuity from samples. The exact result, retaining the requested order, is (nonbracket, f(x)=2*x, 0, 1, 4, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r02-intermediate-value-theorem-0077/)

Intermediate value theorem · qf-mod47-ivt-discontinuity-boundary · candidate





## 78. A Discontinuous Sign Jump and Failure of Uniqueness Have Different Boundaries

Use boundary mode=nonunique and positive exact scale=3. Give (mode, function_description, a, b, target, inclusive_bracket, multiplicity_statement, uniqueness_claim).

The three modes have8/7/8 fields: jump has no root, nonbracket fails the bracket, and nonunique has two endpoint roots. The final No has different roles; arity cannot choose the branch.

View answer and explanation(nonunique, f(x)=x^2, -3, 3, 9, Yes, two_roots_possible, No)

The discontinuous jump takes only ±scale and misses0 despite bracketing it on [−1,1]. A scaled identity on [0,1] misses target2*scale, so the bracket fails. For x^2 on [−scale,scale], target scale^2 has both endpoint roots: existence is not uniqueness. Preserve each branch field meaning and do not infer continuity from samples. The exact result, retaining the requested order, is (nonunique, f(x)=x^2, -3, 3, 9, Yes, two_roots_possible, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r02-intermediate-value-theorem-0078/)

Intermediate value theorem · qf-mod47-ivt-discontinuity-boundary · candidate





## 79. A Discontinuous Sign Jump and Failure of Uniqueness Have Different Boundaries

Use boundary mode=discontinuous_jump and positive exact scale=4. Give (mode, function_description, a, b, target, inclusive_bracket, continuous, root_exists).

The three modes have8/7/8 fields: jump has no root, nonbracket fails the bracket, and nonunique has two endpoint roots. The final No has different roles; arity cannot choose the branch.

View answer and explanation(discontinuous_jump, f(x)=-4_for_x<0_and_4_for_x>=0, -1, 1, 0, Yes, No, No)

The discontinuous jump takes only ±scale and misses0 despite bracketing it on [−1,1]. A scaled identity on [0,1] misses target2*scale, so the bracket fails. For x^2 on [−scale,scale], target scale^2 has both endpoint roots: existence is not uniqueness. Preserve each branch field meaning and do not infer continuity from samples. The exact result, retaining the requested order, is (discontinuous_jump, f(x)=-4_for_x<0_and_4_for_x>=0, -1, 1, 0, Yes, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r02-intermediate-value-theorem-0079/)

Intermediate value theorem · qf-mod47-ivt-discontinuity-boundary · candidate





## 80. A Discontinuous Sign Jump and Failure of Uniqueness Have Different Boundaries

Use boundary mode=nonbracket and positive exact scale=5. Give (mode, function_description, a, b, target, inclusive_bracket, root_exists).

The three modes have8/7/8 fields: jump has no root, nonbracket fails the bracket, and nonunique has two endpoint roots. The final No has different roles; arity cannot choose the branch.

View answer and explanation(nonbracket, f(x)=5*x, 0, 1, 10, No, No)

The discontinuous jump takes only ±scale and misses0 despite bracketing it on [−1,1]. A scaled identity on [0,1] misses target2*scale, so the bracket fails. For x^2 on [−scale,scale], target scale^2 has both endpoint roots: existence is not uniqueness. Preserve each branch field meaning and do not infer continuity from samples. The exact result, retaining the requested order, is (nonbracket, f(x)=5*x, 0, 1, 10, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r02-intermediate-value-theorem-0080/)

Intermediate value theorem · qf-mod47-ivt-discontinuity-boundary · candidate





## 81. A Discontinuous Sign Jump and Failure of Uniqueness Have Different Boundaries

Use boundary mode=nonunique and positive exact scale=6. Give (mode, function_description, a, b, target, inclusive_bracket, multiplicity_statement, uniqueness_claim).

The three modes have8/7/8 fields: jump has no root, nonbracket fails the bracket, and nonunique has two endpoint roots. The final No has different roles; arity cannot choose the branch.

View answer and explanation(nonunique, f(x)=x^2, -6, 6, 36, Yes, two_roots_possible, No)

The discontinuous jump takes only ±scale and misses0 despite bracketing it on [−1,1]. A scaled identity on [0,1] misses target2*scale, so the bracket fails. For x^2 on [−scale,scale], target scale^2 has both endpoint roots: existence is not uniqueness. Preserve each branch field meaning and do not infer continuity from samples. The exact result, retaining the requested order, is (nonunique, f(x)=x^2, -6, 6, 36, Yes, two_roots_possible, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r02-intermediate-value-theorem-0081/)

Intermediate value theorem · qf-mod47-ivt-discontinuity-boundary · candidate





## 82. A Discontinuous Sign Jump and Failure of Uniqueness Have Different Boundaries

Use boundary mode=discontinuous_jump and positive exact scale=7. Give (mode, function_description, a, b, target, inclusive_bracket, continuous, root_exists).

The three modes have8/7/8 fields: jump has no root, nonbracket fails the bracket, and nonunique has two endpoint roots. The final No has different roles; arity cannot choose the branch.

View answer and explanation(discontinuous_jump, f(x)=-7_for_x<0_and_7_for_x>=0, -1, 1, 0, Yes, No, No)

The discontinuous jump takes only ±scale and misses0 despite bracketing it on [−1,1]. A scaled identity on [0,1] misses target2*scale, so the bracket fails. For x^2 on [−scale,scale], target scale^2 has both endpoint roots: existence is not uniqueness. Preserve each branch field meaning and do not infer continuity from samples. The exact result, retaining the requested order, is (discontinuous_jump, f(x)=-7_for_x<0_and_7_for_x>=0, -1, 1, 0, Yes, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r02-intermediate-value-theorem-0082/)

Intermediate value theorem · qf-mod47-ivt-discontinuity-boundary · candidate





## 83. A Discontinuous Sign Jump and Failure of Uniqueness Have Different Boundaries

Use boundary mode=nonbracket and positive exact scale=8. Give (mode, function_description, a, b, target, inclusive_bracket, root_exists).

The three modes have8/7/8 fields: jump has no root, nonbracket fails the bracket, and nonunique has two endpoint roots. The final No has different roles; arity cannot choose the branch.

View answer and explanation(nonbracket, f(x)=8*x, 0, 1, 16, No, No)

The discontinuous jump takes only ±scale and misses0 despite bracketing it on [−1,1]. A scaled identity on [0,1] misses target2*scale, so the bracket fails. For x^2 on [−scale,scale], target scale^2 has both endpoint roots: existence is not uniqueness. Preserve each branch field meaning and do not infer continuity from samples. The exact result, retaining the requested order, is (nonbracket, f(x)=8*x, 0, 1, 16, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r02-intermediate-value-theorem-0083/)

Intermediate value theorem · qf-mod47-ivt-discontinuity-boundary · candidate





## 84. A Discontinuous Sign Jump and Failure of Uniqueness Have Different Boundaries

Use boundary mode=nonunique and positive exact scale=9. Give (mode, function_description, a, b, target, inclusive_bracket, multiplicity_statement, uniqueness_claim).

The three modes have8/7/8 fields: jump has no root, nonbracket fails the bracket, and nonunique has two endpoint roots. The final No has different roles; arity cannot choose the branch.

View answer and explanation(nonunique, f(x)=x^2, -9, 9, 81, Yes, two_roots_possible, No)

The discontinuous jump takes only ±scale and misses0 despite bracketing it on [−1,1]. A scaled identity on [0,1] misses target2*scale, so the bracket fails. For x^2 on [−scale,scale], target scale^2 has both endpoint roots: existence is not uniqueness. Preserve each branch field meaning and do not infer continuity from samples. The exact result, retaining the requested order, is (nonunique, f(x)=x^2, -9, 9, 81, Yes, two_roots_possible, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r02-intermediate-value-theorem-0084/)

Intermediate value theorem · qf-mod47-ivt-discontinuity-boundary · candidate





## 85. A Discontinuous Sign Jump and Failure of Uniqueness Have Different Boundaries

Use boundary mode=discontinuous_jump and positive exact scale=10. Give (mode, function_description, a, b, target, inclusive_bracket, continuous, root_exists).

The three modes have8/7/8 fields: jump has no root, nonbracket fails the bracket, and nonunique has two endpoint roots. The final No has different roles; arity cannot choose the branch.

View answer and explanation(discontinuous_jump, f(x)=-10_for_x<0_and_10_for_x>=0, -1, 1, 0, Yes, No, No)

The discontinuous jump takes only ±scale and misses0 despite bracketing it on [−1,1]. A scaled identity on [0,1] misses target2*scale, so the bracket fails. For x^2 on [−scale,scale], target scale^2 has both endpoint roots: existence is not uniqueness. Preserve each branch field meaning and do not infer continuity from samples. The exact result, retaining the requested order, is (discontinuous_jump, f(x)=-10_for_x<0_and_10_for_x>=0, -1, 1, 0, Yes, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r02-intermediate-value-theorem-0085/)

Intermediate value theorem · qf-mod47-ivt-discontinuity-boundary · candidate





## 86. A Discontinuous Sign Jump and Failure of Uniqueness Have Different Boundaries

Use boundary mode=nonbracket and positive exact scale=11. Give (mode, function_description, a, b, target, inclusive_bracket, root_exists).

The three modes have8/7/8 fields: jump has no root, nonbracket fails the bracket, and nonunique has two endpoint roots. The final No has different roles; arity cannot choose the branch.

View answer and explanation(nonbracket, f(x)=11*x, 0, 1, 22, No, No)

The discontinuous jump takes only ±scale and misses0 despite bracketing it on [−1,1]. A scaled identity on [0,1] misses target2*scale, so the bracket fails. For x^2 on [−scale,scale], target scale^2 has both endpoint roots: existence is not uniqueness. Preserve each branch field meaning and do not infer continuity from samples. The exact result, retaining the requested order, is (nonbracket, f(x)=11*x, 0, 1, 22, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r02-intermediate-value-theorem-0086/)

Intermediate value theorem · qf-mod47-ivt-discontinuity-boundary · candidate





## 87. A Discontinuous Sign Jump and Failure of Uniqueness Have Different Boundaries

Use boundary mode=nonunique and positive exact scale=12. Give (mode, function_description, a, b, target, inclusive_bracket, multiplicity_statement, uniqueness_claim).

The three modes have8/7/8 fields: jump has no root, nonbracket fails the bracket, and nonunique has two endpoint roots. The final No has different roles; arity cannot choose the branch.

View answer and explanation(nonunique, f(x)=x^2, -12, 12, 144, Yes, two_roots_possible, No)

The discontinuous jump takes only ±scale and misses0 despite bracketing it on [−1,1]. A scaled identity on [0,1] misses target2*scale, so the bracket fails. For x^2 on [−scale,scale], target scale^2 has both endpoint roots: existence is not uniqueness. Preserve each branch field meaning and do not infer continuity from samples. The exact result, retaining the requested order, is (nonunique, f(x)=x^2, -12, 12, 144, Yes, two_roots_possible, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r02-intermediate-value-theorem-0087/)

Intermediate value theorem · qf-mod47-ivt-discontinuity-boundary · candidate





## 88. A Discontinuous Sign Jump and Failure of Uniqueness Have Different Boundaries

Use boundary mode=discontinuous_jump and positive exact scale=13. Give (mode, function_description, a, b, target, inclusive_bracket, continuous, root_exists).

The three modes have8/7/8 fields: jump has no root, nonbracket fails the bracket, and nonunique has two endpoint roots. The final No has different roles; arity cannot choose the branch.

View answer and explanation(discontinuous_jump, f(x)=-13_for_x<0_and_13_for_x>=0, -1, 1, 0, Yes, No, No)

The discontinuous jump takes only ±scale and misses0 despite bracketing it on [−1,1]. A scaled identity on [0,1] misses target2*scale, so the bracket fails. For x^2 on [−scale,scale], target scale^2 has both endpoint roots: existence is not uniqueness. Preserve each branch field meaning and do not infer continuity from samples. The exact result, retaining the requested order, is (discontinuous_jump, f(x)=-13_for_x<0_and_13_for_x>=0, -1, 1, 0, Yes, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r02-intermediate-value-theorem-0088/)

Intermediate value theorem · qf-mod47-ivt-discontinuity-boundary · candidate





## 89. A Discontinuous Sign Jump and Failure of Uniqueness Have Different Boundaries

Use boundary mode=nonbracket and positive exact scale=14. Give (mode, function_description, a, b, target, inclusive_bracket, root_exists).

The three modes have8/7/8 fields: jump has no root, nonbracket fails the bracket, and nonunique has two endpoint roots. The final No has different roles; arity cannot choose the branch.

View answer and explanation(nonbracket, f(x)=14*x, 0, 1, 28, No, No)

The discontinuous jump takes only ±scale and misses0 despite bracketing it on [−1,1]. A scaled identity on [0,1] misses target2*scale, so the bracket fails. For x^2 on [−scale,scale], target scale^2 has both endpoint roots: existence is not uniqueness. Preserve each branch field meaning and do not infer continuity from samples. The exact result, retaining the requested order, is (nonbracket, f(x)=14*x, 0, 1, 28, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r02-intermediate-value-theorem-0089/)

Intermediate value theorem · qf-mod47-ivt-discontinuity-boundary · candidate





## 90. A Discontinuous Sign Jump and Failure of Uniqueness Have Different Boundaries

Use boundary mode=nonunique and positive exact scale=15. Give (mode, function_description, a, b, target, inclusive_bracket, multiplicity_statement, uniqueness_claim).

The three modes have8/7/8 fields: jump has no root, nonbracket fails the bracket, and nonunique has two endpoint roots. The final No has different roles; arity cannot choose the branch.

View answer and explanation(nonunique, f(x)=x^2, -15, 15, 225, Yes, two_roots_possible, No)

The discontinuous jump takes only ±scale and misses0 despite bracketing it on [−1,1]. A scaled identity on [0,1] misses target2*scale, so the bracket fails. For x^2 on [−scale,scale], target scale^2 has both endpoint roots: existence is not uniqueness. Preserve each branch field meaning and do not infer continuity from samples. The exact result, retaining the requested order, is (nonunique, f(x)=x^2, -15, 15, 225, Yes, two_roots_possible, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r02-intermediate-value-theorem-0090/)

Intermediate value theorem · qf-mod47-ivt-discontinuity-boundary · candidate





## 91. A Discontinuous Sign Jump and Failure of Uniqueness Have Different Boundaries

Use boundary mode=discontinuous_jump and positive exact scale=16. Give (mode, function_description, a, b, target, inclusive_bracket, continuous, root_exists).

The three modes have8/7/8 fields: jump has no root, nonbracket fails the bracket, and nonunique has two endpoint roots. The final No has different roles; arity cannot choose the branch.

View answer and explanation(discontinuous_jump, f(x)=-16_for_x<0_and_16_for_x>=0, -1, 1, 0, Yes, No, No)

The discontinuous jump takes only ±scale and misses0 despite bracketing it on [−1,1]. A scaled identity on [0,1] misses target2*scale, so the bracket fails. For x^2 on [−scale,scale], target scale^2 has both endpoint roots: existence is not uniqueness. Preserve each branch field meaning and do not infer continuity from samples. The exact result, retaining the requested order, is (discontinuous_jump, f(x)=-16_for_x<0_and_16_for_x>=0, -1, 1, 0, Yes, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r02-intermediate-value-theorem-0091/)

Intermediate value theorem · qf-mod47-ivt-discontinuity-boundary · candidate





## 92. A Discontinuous Sign Jump and Failure of Uniqueness Have Different Boundaries

Use boundary mode=nonbracket and positive exact scale=17. Give (mode, function_description, a, b, target, inclusive_bracket, root_exists).

The three modes have8/7/8 fields: jump has no root, nonbracket fails the bracket, and nonunique has two endpoint roots. The final No has different roles; arity cannot choose the branch.

View answer and explanation(nonbracket, f(x)=17*x, 0, 1, 34, No, No)

The discontinuous jump takes only ±scale and misses0 despite bracketing it on [−1,1]. A scaled identity on [0,1] misses target2*scale, so the bracket fails. For x^2 on [−scale,scale], target scale^2 has both endpoint roots: existence is not uniqueness. Preserve each branch field meaning and do not infer continuity from samples. The exact result, retaining the requested order, is (nonbracket, f(x)=17*x, 0, 1, 34, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r02-intermediate-value-theorem-0092/)

Intermediate value theorem · qf-mod47-ivt-discontinuity-boundary · candidate





## 93. A Discontinuous Sign Jump and Failure of Uniqueness Have Different Boundaries

Use boundary mode=nonunique and positive exact scale=18. Give (mode, function_description, a, b, target, inclusive_bracket, multiplicity_statement, uniqueness_claim).

The three modes have8/7/8 fields: jump has no root, nonbracket fails the bracket, and nonunique has two endpoint roots. The final No has different roles; arity cannot choose the branch.

View answer and explanation(nonunique, f(x)=x^2, -18, 18, 324, Yes, two_roots_possible, No)

The discontinuous jump takes only ±scale and misses0 despite bracketing it on [−1,1]. A scaled identity on [0,1] misses target2*scale, so the bracket fails. For x^2 on [−scale,scale], target scale^2 has both endpoint roots: existence is not uniqueness. Preserve each branch field meaning and do not infer continuity from samples. The exact result, retaining the requested order, is (nonunique, f(x)=x^2, -18, 18, 324, Yes, two_roots_possible, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r02-intermediate-value-theorem-0093/)

Intermediate value theorem · qf-mod47-ivt-discontinuity-boundary · candidate





## 94. A Discontinuous Sign Jump and Failure of Uniqueness Have Different Boundaries

Use boundary mode=discontinuous_jump and positive exact scale=19. Give (mode, function_description, a, b, target, inclusive_bracket, continuous, root_exists).

The three modes have8/7/8 fields: jump has no root, nonbracket fails the bracket, and nonunique has two endpoint roots. The final No has different roles; arity cannot choose the branch.

View answer and explanation(discontinuous_jump, f(x)=-19_for_x<0_and_19_for_x>=0, -1, 1, 0, Yes, No, No)

The discontinuous jump takes only ±scale and misses0 despite bracketing it on [−1,1]. A scaled identity on [0,1] misses target2*scale, so the bracket fails. For x^2 on [−scale,scale], target scale^2 has both endpoint roots: existence is not uniqueness. Preserve each branch field meaning and do not infer continuity from samples. The exact result, retaining the requested order, is (discontinuous_jump, f(x)=-19_for_x<0_and_19_for_x>=0, -1, 1, 0, Yes, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r02-intermediate-value-theorem-0094/)

Intermediate value theorem · qf-mod47-ivt-discontinuity-boundary · candidate





## 95. A Discontinuous Sign Jump and Failure of Uniqueness Have Different Boundaries

Use boundary mode=nonbracket and positive exact scale=20. Give (mode, function_description, a, b, target, inclusive_bracket, root_exists).

The three modes have8/7/8 fields: jump has no root, nonbracket fails the bracket, and nonunique has two endpoint roots. The final No has different roles; arity cannot choose the branch.

View answer and explanation(nonbracket, f(x)=20*x, 0, 1, 40, No, No)

The discontinuous jump takes only ±scale and misses0 despite bracketing it on [−1,1]. A scaled identity on [0,1] misses target2*scale, so the bracket fails. For x^2 on [−scale,scale], target scale^2 has both endpoint roots: existence is not uniqueness. Preserve each branch field meaning and do not infer continuity from samples. The exact result, retaining the requested order, is (nonbracket, f(x)=20*x, 0, 1, 40, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r02-intermediate-value-theorem-0095/)

Intermediate value theorem · qf-mod47-ivt-discontinuity-boundary · candidate





## 96. A Discontinuous Sign Jump and Failure of Uniqueness Have Different Boundaries

Use boundary mode=nonunique and positive exact scale=21. Give (mode, function_description, a, b, target, inclusive_bracket, multiplicity_statement, uniqueness_claim).

The three modes have8/7/8 fields: jump has no root, nonbracket fails the bracket, and nonunique has two endpoint roots. The final No has different roles; arity cannot choose the branch.

View answer and explanation(nonunique, f(x)=x^2, -21, 21, 441, Yes, two_roots_possible, No)

The discontinuous jump takes only ±scale and misses0 despite bracketing it on [−1,1]. A scaled identity on [0,1] misses target2*scale, so the bracket fails. For x^2 on [−scale,scale], target scale^2 has both endpoint roots: existence is not uniqueness. Preserve each branch field meaning and do not infer continuity from samples. The exact result, retaining the requested order, is (nonunique, f(x)=x^2, -21, 21, 441, Yes, two_roots_possible, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r02-intermediate-value-theorem-0096/)

Intermediate value theorem · qf-mod47-ivt-discontinuity-boundary · candidate





## 97. A Discontinuous Sign Jump and Failure of Uniqueness Have Different Boundaries

Use boundary mode=discontinuous_jump and positive exact scale=22. Give (mode, function_description, a, b, target, inclusive_bracket, continuous, root_exists).

The three modes have8/7/8 fields: jump has no root, nonbracket fails the bracket, and nonunique has two endpoint roots. The final No has different roles; arity cannot choose the branch.

View answer and explanation(discontinuous_jump, f(x)=-22_for_x<0_and_22_for_x>=0, -1, 1, 0, Yes, No, No)

The discontinuous jump takes only ±scale and misses0 despite bracketing it on [−1,1]. A scaled identity on [0,1] misses target2*scale, so the bracket fails. For x^2 on [−scale,scale], target scale^2 has both endpoint roots: existence is not uniqueness. Preserve each branch field meaning and do not infer continuity from samples. The exact result, retaining the requested order, is (discontinuous_jump, f(x)=-22_for_x<0_and_22_for_x>=0, -1, 1, 0, Yes, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r02-intermediate-value-theorem-0097/)

Intermediate value theorem · qf-mod47-ivt-discontinuity-boundary · candidate





## 98. A Discontinuous Sign Jump and Failure of Uniqueness Have Different Boundaries

Use boundary mode=nonbracket and positive exact scale=23. Give (mode, function_description, a, b, target, inclusive_bracket, root_exists).

The three modes have8/7/8 fields: jump has no root, nonbracket fails the bracket, and nonunique has two endpoint roots. The final No has different roles; arity cannot choose the branch.

View answer and explanation(nonbracket, f(x)=23*x, 0, 1, 46, No, No)

The discontinuous jump takes only ±scale and misses0 despite bracketing it on [−1,1]. A scaled identity on [0,1] misses target2*scale, so the bracket fails. For x^2 on [−scale,scale], target scale^2 has both endpoint roots: existence is not uniqueness. Preserve each branch field meaning and do not infer continuity from samples. The exact result, retaining the requested order, is (nonbracket, f(x)=23*x, 0, 1, 46, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r02-intermediate-value-theorem-0098/)

Intermediate value theorem · qf-mod47-ivt-discontinuity-boundary · candidate





## 99. A Discontinuous Sign Jump and Failure of Uniqueness Have Different Boundaries

Use boundary mode=nonunique and positive exact scale=24. Give (mode, function_description, a, b, target, inclusive_bracket, multiplicity_statement, uniqueness_claim).

The three modes have8/7/8 fields: jump has no root, nonbracket fails the bracket, and nonunique has two endpoint roots. The final No has different roles; arity cannot choose the branch.

View answer and explanation(nonunique, f(x)=x^2, -24, 24, 576, Yes, two_roots_possible, No)

The discontinuous jump takes only ±scale and misses0 despite bracketing it on [−1,1]. A scaled identity on [0,1] misses target2*scale, so the bracket fails. For x^2 on [−scale,scale], target scale^2 has both endpoint roots: existence is not uniqueness. Preserve each branch field meaning and do not infer continuity from samples. The exact result, retaining the requested order, is (nonunique, f(x)=x^2, -24, 24, 576, Yes, two_roots_possible, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r02-intermediate-value-theorem-0099/)

Intermediate value theorem · qf-mod47-ivt-discontinuity-boundary · candidate





## 100. A Discontinuous Sign Jump and Failure of Uniqueness Have Different Boundaries

Use boundary mode=discontinuous_jump and positive exact scale=25. Give (mode, function_description, a, b, target, inclusive_bracket, continuous, root_exists).

The three modes have8/7/8 fields: jump has no root, nonbracket fails the bracket, and nonunique has two endpoint roots. The final No has different roles; arity cannot choose the branch.

View answer and explanation(discontinuous_jump, f(x)=-25_for_x<0_and_25_for_x>=0, -1, 1, 0, Yes, No, No)

The discontinuous jump takes only ±scale and misses0 despite bracketing it on [−1,1]. A scaled identity on [0,1] misses target2*scale, so the bracket fails. For x^2 on [−scale,scale], target scale^2 has both endpoint roots: existence is not uniqueness. Preserve each branch field meaning and do not infer continuity from samples. The exact result, retaining the requested order, is (discontinuous_jump, f(x)=-25_for_x<0_and_25_for_x>=0, -1, 1, 0, Yes, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r02-intermediate-value-theorem-0100/)

Intermediate value theorem · qf-mod47-ivt-discontinuity-boundary · candidate



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