# Question 22215: Failed MVT Hypotheses Need Their Own Exact Output Layout

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This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

R0043-22215 · q-20261014-advance-20261110-r02-mean-value-theorem-0090



## Original question and givens

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


- G001 — Use boundary mode=cusp_at_zero, a=-15, b=16, scale=15, offset=14. degenerate_interval declares a≥b; jump_at_zero and cusp_at_zero declare a<0<b and nonzero scale, with the named interior defect. Give (mode, a, b, scale, offset, closed_interval_continuity, open_interval_differentiability, theorem_applicability, boundary).





## Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.


- [mko-mean-value-theorem](https://ocme.evemisslab.com/en/mko/mko-mean-value-theorem/)





## Step-by-step derivation

S001 · impliesBefore

source operation=mvt-failure-boundary;original snapshot and mathematics retained

After

explicit ordered source output fields=[mode, a, b, scale, offset, closed_interval_continuity, open_interval_differentiability, theorem_applicability, boundary]

Conditions: none

Retain the historical interface marker and list ordered source fields without JavaScript.MVT fields come from the original mode,without padding or answer-shape inference.Tags and fixed No markers retain their claim scope.

Step identifiers and dependenciesRule: source_ordered_interface_explicit_without_javascript · Uses: G001





S002 · impliesBefore

ORIGINAL mode=cusp_at_zero;a=-15<0<b=16;nonzero scale=15;offset=14

After

closed-interval continuity=Yes;open-interval differentiability=No;theorem applicability=No;ORIGINAL9-field layout

Conditions: none

Zero is interior.A nonzero jump has different one-sided limits and is neither continuous nor differentiable.A cusp has opposite nonzero one-sided slopes:continuous but not differentiable.Adding an offset removes neither defect.The No withholds MVT application under these failed hypotheses.

Step identifiers and dependenciesRule: mvt_actual_interior_jump_or_cusp_nonzero_scale · Uses: G001, S001









## Reverse checks


- S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.





## Result

(cusp_at_zero, -15, 16, 15, 14, Yes, No, No, open_interval_differentiability_fails_at_zero)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.





## This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.

Source hashes and review receipts
- input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
- original question SHA-256: cc2e39affd4aa71145bd3b008aaa1a4d89e8c993af4d9bd2473c4082c4a4a56c
- candidate record SHA-256: a79a73c28b9fedddfe2b5654ef3b25c82eb41659e1d84bf92f870139493b781e
- semantic review receipt SHA-256: 49cf6ec60211c92d1c0c302663af4d44479635dc8181fb61a20c3374288e21c5
- question-first witness SHA-256: 10fd25b4643e5ff12753daceaa61f9f2499bf381f65ef96e9d21185c7c0d0c5e
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261110-r02-mean-value-theorem-0090.json)