# 第 22247 題：微分中值定理

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本頁是已完成第二對話 AI 數理與中英語義覆核的教學詳解，只支持本題明示的模型與前提。沒有人工審查或新 Lean 證明，也不提升原題或 MKO 的 Canonical 真值狀態。

R0043-22247 · q-20261014-advance-20261110-r02-mean-value-theorem-0022



## 原題與已知條件

f(t)=A*t^2+B*t+C、a<b、closed interval continuity與open interval differentiability。操作=mvt-hypothesis-audit；資料={"A":2,"B":11,"C":442,"a":-23,"b":24}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。


- G001 — f(t)=A*t^2+B*t+C、a<b、closed interval continuity與open interval differentiability。操作=mvt-hypothesis-audit；資料={"A":2,"B":11,"C":442,"a":-23,"b":24}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。





## 定義與適用範圍

沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。


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





## 逐步推導

S001 · implies操作前

source operation=mvt-hypothesis-audit;original snapshot and mathematics retained

操作後

explicit ordered source output fields=[a, b, ordered_endpoints, closed_interval_continuity, open_interval_differentiability, certificate]

條件：無

原題歷史介面標記保留，直接列來源有序欄位，不需JavaScript；MVT由原mode選欄序，不補空欄或看答案形狀猜。tag與固定No保原聲稱範圍。

步驟識別與依賴規則：source_ordered_interface_explicit_without_javascript · 依賴：G001





S002 · implies操作前

ORIGINAL WHOLE f(t)=(2)t^2+(11)t+(442);a=-23<b=24;all endpoint[Ax,Axx,Bx,Axx+Bx,Axx+Bx+C] rows=[[-46, 1058, -253, 805, 1247], [48, 1152, 264, 1416, 1858]]

操作後

English source ALL endpoint intermediate SAFE guards passed;polynomial continuous on CLOSED[a,b] and differentiable on OPEN(a,b),not inferred from finite samples

條件：無

有限和／乘積保連續與可微，導數是2At+B；這是完整公式的性質，不是用端點樣本猜中間。SAFE只是來源數字投影資格，不等於數學上多項式只能取這些值。

步驟識別與依賴規則：mvt_whole_polynomial_hypotheses_and_SAFE_source_guard · 依賴：G001, S001









## 逐步反向核對


- S002 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
- S001 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。





## 結論

(-23, 24, 是, 是, 是, polynomial_continuous_on_closed_interval_and_differentiable_on_open_interval)

依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。





## 本題覆核與來源綁定

覆核日期：2026-10-26

原題、答案、步驟表達式、數值 AST 與既有歷史保持原樣；原來的候選狀態屬於來源歷史，不覆寫為形式化證明。

來源雜湊與覆核憑證
- input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
- original question SHA-256: cfc8626e22ed8b5f300bbf0a574d858108f8aba290703a27569caccdab4653af
- candidate record SHA-256: 5bc604a37f20a288a73418fbabb8aae4cdbb4d419d4661c2a8d70285018d3ff4
- semantic review receipt SHA-256: 49cf6ec60211c92d1c0c302663af4d44479635dc8181fb61a20c3374288e21c5
- question-first witness SHA-256: 10fd25b4643e5ff12753daceaa61f9f2499bf381f65ef96e9d21185c7c0d0c5e
- Lean: NOT_RUN · human review: NOT_PERFORMED

[同源完整 JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261110-r02-mean-value-theorem-0022.json)