第 22221 題：微分中值定理

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-failure-boundary；資料={"mode":"cusp_at_zero","a":-21,"b":22,"scale":21,"offset":20}。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-failure-boundary；資料={"mode":"cusp_at_zero","a":-21,"b":22,"scale":21,"offset":20}。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

逐步推導

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

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

ORIGINAL mode=cusp_at_zero;a=-21<0<b=22;nonzero scale=21;offset=20

closed-interval continuity=是;open-interval differentiability=否;theorem applicability=否;ORIGINAL9-field layout

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

(cusp_at_zero, -21, 22, 21, 20, 是, 否, 否, open_interval_differentiability_fails_at_zero)

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。