{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261007-advance-20261009-r04-derivative-rules-0001",
  "pilot_number": "R0006-03651",
  "number": 3651,
  "family_id": "qf-density-derivative-second-value",
  "title": {
    "zh-Hant": "第 3651 題：多項式的二階點導數",
    "en": "Question 3651: Second Derivative of a Polynomial at a Point"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261007-advance-20261009-r04-derivative-rules-0001",
    "batch_id": "batch-20261007-advance-20261009-r04-derivative-rules-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-derivative-rules"
    ],
    "title_zh": "多項式的二階點導數",
    "stem_zh": "全實數論域；升冪係數的受限多項式模型。升冪係數 f=[0]，求 f″(2)，採最簡正分母分數。",
    "answer": {
      "kind": "expression",
      "value": "0/1",
      "explanation_zh": "二階導數=0/1。產生端Horner同步傳遞值、一階及二階導數；常數及線性式二階導數為零，不以有限點抽樣當恆等證明。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 1,
      "skills": [
        "多項式的二階點導數"
      ]
    },
    "variant": {
      "family_id": "qf-density-derivative-second-value",
      "seed": 202610094001,
      "parameters": {
        "f": [
          0
        ],
        "x": 2,
        "operation": "density_derivative_second_value",
        "domain": "reals",
        "representation": "ascending_powers",
        "semantic_template": "qf-density-derivative-second-value/[\"zero_second_derivative\"]"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_integer_rational_arithmetic",
        "producer_finite_enumeration",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-07",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261007-advance-20261009-r04-derivative-rules-0001",
    "batch_id": "batch-20261007-advance-20261009-r04-derivative-rules-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "6669b46625ec87b01307a396ab8822ab1d31b78239e0e708cbc6fb1ac807a51b",
      "family_id": "qf-density-derivative-second-value",
      "seed": 202610094001,
      "semantic_template": "qf-density-derivative-second-value/[\"zero_second_derivative\"]"
    },
    "fields": {
      "title": "Second Derivative of a Polynomial at a Point",
      "stem": "On all R, f has ascending coefficients [0], constant term first. Find f″(2). Use reduced positive-denominator fractions, retaining /1 for integers.",
      "answer": {
        "kind": "expression",
        "value": "0/1",
        "explanation": "Differentiate twice and evaluate exactly, obtaining 0/1. Constant and linear inputs have zero second derivative; this declared polynomial calculation is not sampling of arbitrary functions."
      },
      "skills": [
        "Second Derivative of a Polynomial at a Point"
      ],
      "verification_warning": "This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."
    }
  },
  "givens": [
    {
      "id": "G001",
      "text": {
        "zh-Hant": "全實數論域；升冪係數的受限多項式模型。升冪係數 f=[0]，求 f″(2)，採最簡正分母分數。",
        "en": "On all R, f has ascending coefficients [0], constant term first. Find f″(2). Use reduced positive-denominator fractions, retaining /1 for integers."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "f ascending coefficients=[0]",
      "after": "real polynomials are differentiable;use coefficient identities",
      "relation": "implies",
      "rule_id": "polynomial_derivative_premise",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第k個係數乘x^k；實多項式可逐項微分，常數導數0。公式恆等與只在一點比較候選是不同主張。",
        "en": "Coefficient k multiplies x^k;real polynomials differentiate termwise and constants have derivative0.Coefficient identities and a candidate comparison at one point are different claims."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "f″(x) ascending coefficients=[0]",
      "after": "evaluate at x=2",
      "relation": "implies",
      "rule_id": "polynomial_evaluation_order",
      "conditions": [],
      "explanation": {
        "zh-Hant": "升冪係數用Horner由最高非零項向常數項求值；略去零最高項不改函數，也不改原始係數清單。",
        "en": "Evaluate ascending coefficients by Horner from the highest nonzero term down;ignoring redundant top zero terms changes neither the function nor the retained original list."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "first coefficients=[0];second=[0]",
      "after": "f″(2)=0",
      "relation": "implies",
      "rule_id": "successive_polynomial_derivatives",
      "conditions": [],
      "explanation": {
        "zh-Hant": "微分兩次，每次用當前次數；常數與線性式的二階導數0，不把一階導數再平方。",
        "en": "Differentiate twice using each current degree;constants and linear polynomials have second derivative0.Do not square the first derivative."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "result": "0",
    "first_derivative": [
      "0"
    ],
    "second_derivative": [
      "0"
    ]
  },
  "conclusion": {
    "canonical_value": "0/1",
    "explanation": {
      "zh-Hant": "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。",
      "en": "The full substitutions and premise checks give the original answer; source identity,format and status are unchanged."
    }
  },
  "definition_note": {
    "zh-Hant": "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。",
    "en": "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."
  },
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      "question_compact_sha256": "6669b46625ec87b01307a396ab8822ab1d31b78239e0e708cbc6fb1ac807a51b",
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      "english_rows_path": "public/data/i18n/questions/batch-20261007-advance-20261009-r04-derivative-rules-foundations.en.jsonl"
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        "original_path": "public/data/mko/derivative-rules.json",
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    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "6bac5f83f28cdd9c992a469306bfdf5c5415de17d11847cb9b17b4d13934c7e6",
    "original_candidate_sha256": "9412da7d4dcef674165b6327e0ccdd3c25b411610988d87706a9cf624c9038d2",
    "original_candidate_line_sha256": "9412da7d4dcef674165b6327e0ccdd3c25b411610988d87706a9cf624c9038d2"
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  "historical_source_state": {
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      "derivation": "candidate_authored_and_locally_checked",
      "source_question_state": "candidate",
      "independent_AI_or_human_review": "NOT_PERFORMED",
      "formalization": "not_formalized",
      "Lean": "NOT_RUN"
    },
    "scope_note": {
      "zh-Hant": "來源綁定教學推導候選；精確核對不等於獨立AI／人工審查或一般形式證明。原題、答案、MKO、公式、AST與歷史不變。",
      "en": "Source-bound pedagogical derivation candidate.Exact checking is not independent AI/human review or general formal proof.Original questions,answers,MKOs,formulas,ASTs and history remain unchanged."
    },
    "label": "Frozen producer metadata; current per-question review is separate below."
  },
  "revision": {
    "id": "review-20261026-004-p7",
    "review_date": "2026-10-26",
    "owner_authorized_reader_publication": true,
    "scope": "One hundred sixty family prototypes and their mathematical/bilingual reasoning variants, stems, boundary statements and every distinct explanation shape were reviewed in a context independent of the producer; parameter substitutions, source bindings, full math models, both answer strings and every arithmetic step were independently reconstructed per question from the original parameters only, with negative controls. This grouped method does not claim a separate literal AI reading of every repeated sentence or an extra AI voter.",
    "review_receipts": {
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    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 3,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
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    "Canonical_truth_promotion": false,
    "authority": "Owner-authorized reviewed reader publication; original question/candidate states remain historical and unchanged."
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}