第 23077 題：高階導數

原題與已知條件

f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-zero-beyond-degree；資料={"coefficients":[2,4,6,8,10],"order":6,"point":-1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

• G001 — f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-zero-beyond-degree；資料={"coefficients":[2,4,6,8,10],"order":6,"point":-1}。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-higher-order-derivative

逐步推導

source operation=higher-zero-beyond-degree;original snapshot and mathematics retained

explicit ordered source output fields=[declared_degree_bound, order, derivative_coefficients, point, value, zero_derivative_certificate, certificate]

ORIGINAL low-to-high coefficients=[2, 4, 6, 8, 10];declared order=6

declared degree BOUND=4=ORIGINAL length-1;initialize derivative order0 with the UNTRIMMED original list

Higher derivative level1 ORIGINAL power1 coefficient product := mul(4, 1)

Higher derivative level1 ORIGINAL power1 coefficient product = 4

Higher derivative level1 ORIGINAL power2 coefficient product := mul(6, 2)

Higher derivative level1 ORIGINAL power2 coefficient product = 12

Higher derivative level1 ORIGINAL power3 coefficient product := mul(8, 3)

Higher derivative level1 ORIGINAL power3 coefficient product = 24

Higher derivative level1 ORIGINAL power4 coefficient product := mul(10, 4)

Higher derivative level1 ORIGINAL power4 coefficient product = 40

previous derivative order=0;actual low-to-high list=[2, 4, 6, 8, 10];drop ONLY its constant term

ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 4, 1, 4], [2, 6, 2, 12], [3, 8, 3, 24], [4, 10, 4, 40]];next raw list=[4, 12, 24, 40];all intermediate products SAFE

Higher derivative level2 ORIGINAL power1 coefficient product := mul(12, 1)

Higher derivative level2 ORIGINAL power1 coefficient product = 12

Higher derivative level2 ORIGINAL power2 coefficient product := mul(24, 2)

Higher derivative level2 ORIGINAL power2 coefficient product = 48

Higher derivative level2 ORIGINAL power3 coefficient product := mul(40, 3)

Higher derivative level2 ORIGINAL power3 coefficient product = 120

previous derivative order=1;actual low-to-high list=[4, 12, 24, 40];drop ONLY its constant term

ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 12, 1, 12], [2, 24, 2, 48], [3, 40, 3, 120]];next raw list=[12, 48, 120];all intermediate products SAFE

Higher derivative level3 ORIGINAL power1 coefficient product := mul(48, 1)

Higher derivative level3 ORIGINAL power1 coefficient product = 48

Higher derivative level3 ORIGINAL power2 coefficient product := mul(120, 2)

Higher derivative level3 ORIGINAL power2 coefficient product = 240

previous derivative order=2;actual low-to-high list=[12, 48, 120];drop ONLY its constant term

ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 48, 1, 48], [2, 120, 2, 240]];next raw list=[48, 240];all intermediate products SAFE

Higher derivative level4 ORIGINAL power1 coefficient product := mul(240, 1)

Higher derivative level4 ORIGINAL power1 coefficient product = 240

previous derivative order=3;actual low-to-high list=[48, 240];drop ONLY its constant term

ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 240, 1, 240]];next raw list=[240];all intermediate products SAFE

previous derivative order=4;actual low-to-high list=[240];drop ONLY its constant term

ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[];next raw list=[];all intermediate products SAFE

previous derivative order=5;actual low-to-high list=[];drop ONLY its constant term

ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[];next raw list=[];all intermediate products SAFE

ORIGINAL coefficients=[2, 4, 6, 8, 10];order=6;declared degree bound=4

final derivative coefficients=[0];for each original power k>=order multiplier=k*(k-1)*...*(k-order+1);lower powers vanish

Higher derivative Horner power0 previous value times ORIGINAL point := mul(0, -1)

Higher derivative Horner power0 previous value times ORIGINAL point = 0

Higher derivative Horner power0 plus ORIGINAL derivative coefficient := sum(0, 0)

Higher derivative Horner power0 plus ORIGINAL derivative coefficient = 0

逐步反向核對

• S021 · premise_and_result_audit — 以精確有理數重算 Higher derivative Horner power0 plus ORIGINAL derivative coefficient，不以小數近似替代。
• S020 · premise_and_result_audit — 以精確有理數重算 Higher derivative Horner power0 previous value times ORIGINAL point，不以小數近似替代。
• S019 · 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。
• S018 · 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。
• S017 · 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。
• S016 · 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。
• S015 · premise_and_result_audit — 以精確有理數重算 Higher derivative level4 ORIGINAL power1 coefficient product，不以小數近似替代。
• S014 · 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。
• S013 · premise_and_result_audit — 以精確有理數重算 Higher derivative level3 ORIGINAL power2 coefficient product，不以小數近似替代。
• S012 · premise_and_result_audit — 以精確有理數重算 Higher derivative level3 ORIGINAL power1 coefficient product，不以小數近似替代。
• S011 · 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。
• S010 · premise_and_result_audit — 以精確有理數重算 Higher derivative level2 ORIGINAL power3 coefficient product，不以小數近似替代。
• S009 · premise_and_result_audit — 以精確有理數重算 Higher derivative level2 ORIGINAL power2 coefficient product，不以小數近似替代。
• S008 · premise_and_result_audit — 以精確有理數重算 Higher derivative level2 ORIGINAL power1 coefficient product，不以小數近似替代。
• S007 · 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。
• S006 · premise_and_result_audit — 以精確有理數重算 Higher derivative level1 ORIGINAL power4 coefficient product，不以小數近似替代。
• S005 · premise_and_result_audit — 以精確有理數重算 Higher derivative level1 ORIGINAL power3 coefficient product，不以小數近似替代。
• S004 · premise_and_result_audit — 以精確有理數重算 Higher derivative level1 ORIGINAL power2 coefficient product，不以小數近似替代。
• S003 · premise_and_result_audit — 以精確有理數重算 Higher derivative level1 ORIGINAL power1 coefficient product，不以小數近似替代。
• 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

(4, 6, [0], -1, 0, 是, orders_above_polynomial_degree_are_zero_without_analyticity_claim)

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。