{
  "schema_version": "ocme-derivation-reader-v1",
  "guide_id": "derivation-mko-mean-value-theorem-01",
  "mko_id": "mko-mean-value-theorem",
  "title": {
    "zh-Hant": "二次多項式的均值點",
    "en": "The mean-value point of a quadratic"
  },
  "contract": {
    "schema_version": "ocme-derivation-guide-v0.1",
    "guide_id": "derivation-mko-mean-value-theorem-01",
    "source": {
      "mko_id": "mko-mean-value-theorem",
      "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
      "sha256": "b56d8d37a21b32c9fbbbb984216cdf4275d9e8a0935e41ffa69680e727899f0b",
      "english_sha256": "5567acf26533764eb5f0ebe7fe5993adffea616c5b5dead898553ab35d8c201e"
    },
    "mode": "worked_example",
    "audience_prerequisites": [
      "algebra",
      "polynomial"
    ],
    "goal": {
      "zh-Hant": "對 f(t)=A t^2+B t+C 在 a<b 上直接構造均值點。",
      "en": "Directly construct a mean-value point for f(t)=A*t^2+B*t+C on a<b."
    },
    "givens": [
      {
        "id": "G1",
        "statement": {
          "zh-Hant": "A,B,C,a,b 為有理數，a<b；f(t)=A t^2+B t+C。",
          "en": "A,B,C,a,b are rational, a<b, and f(t)=A*t^2+B*t+C."
        }
      }
    ],
    "steps": [
      {
        "id": "S01",
        "uses": [
          "G1"
        ],
        "before_tex": "f(t)=At^2+Bt+C",
        "after_tex": "f\\text{ continuous on }[a,b]\\text{ and differentiable on }(a,b)",
        "relation": "implies",
        "rule": "polynomial",
        "conditions": [
          "a<b"
        ],
        "explanation": {
          "zh-Hant": "多項式在全實線連續可微，故兩個不同區間責任均滿足。",
          "en": "A polynomial is continuous and differentiable on R, satisfying both interval-specific hypotheses."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1"
          ],
          "explanation": {
            "zh-Hant": "閉區間連續與開區間可微不能互相替代。",
            "en": "Closed-interval continuity and open-interval differentiability are distinct obligations."
          }
        },
        "leaf_basis": "polynomial"
      },
      {
        "id": "S02",
        "uses": [
          "G1"
        ],
        "before_tex": "f(b)-f(a)",
        "after_tex": "A(b^2-a^2)+B(b-a)=(b-a)(A(a+b)+B)",
        "relation": "equivalent",
        "rule": "algebra",
        "conditions": [],
        "explanation": {
          "zh-Hant": "常數 C 抵銷，用平方差分解。",
          "en": "The constants cancel and the difference of squares factors."
        },
        "reverse_audit": {
          "kind": "inverse_equivalence",
          "required_ids": [
            "G1"
          ],
          "explanation": {
            "zh-Hant": "A/B/C 依序是二次、一次、常數係數，未調換。",
            "en": "A/B/C remain quadratic, linear and constant coefficients in that order."
          }
        },
        "leaf_basis": "algebra"
      },
      {
        "id": "S03",
        "uses": [
          "G1",
          "S02"
        ],
        "before_tex": "(f(b)-f(a))/(b-a)",
        "after_tex": "A(a+b)+B",
        "relation": "equivalent",
        "rule": "algebra",
        "conditions": [
          "b-a>0"
        ],
        "explanation": {
          "zh-Hant": "a<b 保證 b-a>0 可除，得到割線斜率。",
          "en": "a<b gives a positive denominator, yielding the secant slope."
        },
        "reverse_audit": {
          "kind": "inverse_equivalence",
          "required_ids": [
            "G1",
            "S02"
          ],
          "explanation": {
            "zh-Hant": "若 a=b 不能使用這個式子。",
            "en": "This division is unavailable when a=b."
          }
        },
        "leaf_basis": "algebra"
      },
      {
        "id": "S04",
        "uses": [
          "G1"
        ],
        "before_tex": "f(t)=At^2+Bt+C",
        "after_tex": "f'(t)=2At+B",
        "relation": "implies",
        "rule": "polynomial",
        "conditions": [],
        "explanation": {
          "zh-Hant": "逐項微分取得整個開區間上的導函數。",
          "en": "Termwise differentiation gives the derivative on the entire open interval."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1"
          ],
          "explanation": {
            "zh-Hant": "不是只在候選 c 的單點核數值。",
            "en": "This is not only a numerical derivative at a candidate c."
          }
        },
        "leaf_basis": "polynomial"
      },
      {
        "id": "S05",
        "uses": [
          "G1"
        ],
        "before_tex": "c=(a+b)/2",
        "after_tex": "c-a=(b-a)/2>0,\\quad b-c=(b-a)/2>0",
        "relation": "implies",
        "rule": "algebra",
        "conditions": [],
        "explanation": {
          "zh-Hant": "中點嚴格位於兩端之間。",
          "en": "The midpoint lies strictly between both endpoints."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1"
          ],
          "explanation": {
            "zh-Hant": "均值點不允許僅落在端點。",
            "en": "A mean-value witness cannot merely be an endpoint."
          }
        },
        "leaf_basis": "algebra"
      },
      {
        "id": "S06",
        "uses": [
          "S03",
          "S04",
          "S05"
        ],
        "before_tex": "f'(c)=2A(a+b)/2+B",
        "after_tex": "f'(c)=A(a+b)+B=(f(b)-f(a))/(b-a)",
        "relation": "implies",
        "rule": "algebra",
        "conditions": [],
        "explanation": {
          "zh-Hant": "代中點後與割線斜率完全相等，直接證此多項式族的結論。",
          "en": "Substituting the midpoint exactly matches the secant slope, directly proving the conclusion for this polynomial family."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S03",
            "S04",
            "S05"
          ],
          "explanation": {
            "zh-Hant": "等式及內點資格分開已核，沒有把單一數值等式當一般 MVT 證明。",
            "en": "Equality and interior membership were separately checked; a numerical equality is not called a general MVT proof."
          }
        },
        "leaf_basis": "algebra"
      },
      {
        "id": "S07",
        "uses": [
          "S03",
          "S04"
        ],
        "before_tex": "2Ac+B=A(a+b)+B",
        "after_tex": "A\\ne0\\Longrightarrow c=(a+b)/2;\\quad A=0\\Longrightarrow\\forall c\\in(a,b)\\text{ works}",
        "relation": "case_split",
        "rule": "algebra",
        "conditions": [],
        "explanation": {
          "zh-Hant": "非零 A 可約去得唯一中點；仿射與常數情況導數處處等於割線斜率。",
          "en": "Nonzero A gives the unique midpoint; affine and constant functions have every interior point as a witness."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S03",
            "S04"
          ],
          "explanation": {
            "zh-Hant": "存在不普遍推出唯一，A=0 分支不能除 A。",
            "en": "Existence does not imply uniqueness, and division by A is forbidden in the zero branch."
          }
        },
        "leaf_basis": "algebra"
      },
      {
        "id": "S08",
        "uses": [
          "S06"
        ],
        "before_tex": "A=2,B=-3,C=1,a=0,b=2",
        "after_tex": "f(0)=1,\\ f(2)=3,\\ c=1,\\ f'(1)=1=(3-1)/2",
        "relation": "implies",
        "rule": "algebra",
        "conditions": [],
        "explanation": {
          "zh-Hant": "以精確有理實例回查係數與 witness。",
          "en": "An exact rational instance audits coefficient roles and the witness."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S06"
          ],
          "explanation": {
            "zh-Hant": "一般多項式族已在前步證，數值核對只是回查。",
            "en": "The polynomial family was proved earlier; this numerical check is only an audit."
          }
        },
        "leaf_basis": "algebra"
      }
    ],
    "result": {
      "zh-Hant": "中點總是此二次族的均值點；A≠0 時唯一，A=0 時任何內點皆可。",
      "en": "The midpoint is always a mean-value point for this quadratic family; it is unique for A≠0, while every interior point works for A=0."
    },
    "negative_controls": [
      {
        "claim": {
          "zh-Hant": "a=b 仍照除割線分母。",
          "en": "The secant division remains valid for a=b."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "分母為零。",
          "en": "The denominator is zero."
        }
      },
      {
        "claim": {
          "zh-Hant": "|t| 在 [-1,1] 可直接套 MVT。",
          "en": "MVT applies directly to |t| on [-1,1]."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "0 處不可微，且割線斜率 0 無內點導數等於它。",
          "en": "It is not differentiable at zero, and no interior derivative equals the secant slope zero."
        }
      },
      {
        "claim": {
          "zh-Hant": "均值點必唯一。",
          "en": "The mean-value point must be unique."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "A=0 時每個內點都可以。",
          "en": "When A=0 every interior point works."
        }
      }
    ],
    "status": "candidate",
    "gaps": [],
    "verification": {
      "structure": "checked_local_script",
      "mathematics": "candidate_argument",
      "formalization": "not_formalized"
    },
    "rule_glossary": [
      {
        "id": "mko_definition",
        "kind": "cited",
        "statement": {
          "zh-Hant": "題族參數A/B/C明確編碼f(t)=A t^2+B t+C。必須a<b、closed interval continuity與open interval differentiability；witness c嚴格位於(a,b)，且f'(c)=(f(b)-f(a))/(b-a)。c不保證唯一，affine或constant case可有多個witness。有限samples、a=b、jump或interior cusp不足。",
          "en": "Question parameters A/B/C explicitly encode f(t)=A*t^2+B*t+C. Require a<b, continuity on the closed interval and differentiability on the open interval. The witness satisfies a<c<b and f'(c)=(f(b)−f(a))/(b−a). It need not be unique: affine and constant cases can have many witnesses. Finite samples, a=b, a jump or an interior cusp do not supply the required theorem hypotheses."
        },
        "source": {
          "mko_id": "mko-mean-value-theorem",
          "sha256": "b56d8d37a21b32c9fbbbb984216cdf4275d9e8a0935e41ffa69680e727899f0b",
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        }
      },
      {
        "id": "algebra",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "實數域的等式運算；除法前分母必非零，不等式乘正數保持方向。",
          "en": "Use real-field identities; division requires a nonzero denominator and multiplication by a positive number preserves inequalities."
        }
      },
      {
        "id": "polynomial",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "多項式在實數上連續且可逐項微分；d(x^k)/dx=k x^(k-1)，常數導數為零。",
          "en": "Real polynomials are continuous and differentiate termwise: the derivative of x^k is k*x^(k-1), and constants differentiate to zero."
        }
      }
    ],
    "revision_log": [],
    "numeric_checks": [
      {
        "id": "MVT-example",
        "lhs": "4*1-3",
        "rhs": "(3-1)/2",
        "step": "S08"
      }
    ]
  },
  "presentation": {
    "md_scope": {
      "zh-Hant": "精確多項式實例族；不由樣本證一般連續或可微。",
      "en": "An exact polynomial family, not a sampling argument for general continuity or differentiability."
    },
    "md_symbols": [
      [
        "c",
        "嚴格內點均值見證",
        "strict-interior mean-value witness"
      ],
      [
        "A,B,C",
        "依序二次一次常數係數",
        "quadratic, linear, constant coefficients"
      ]
    ],
    "md_error_controls": [
      {
        "title": "分母先行 / Denominator first",
        "zh": "先核 a<b 再除 b-a。",
        "en": "Verify a<b before dividing by b-a."
      },
      {
        "title": "前提分區 / Split interval premises",
        "zh": "全閉區間連續與全開區間可微均由完整多項式給出。",
        "en": "The full polynomial supplies continuity on the whole closed interval and differentiability on the whole interior."
      }
    ],
    "md_open_items": []
  },
  "revision": {
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    "collection": "coverage-20261021",
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    "owner_adopted": true,
    "bilingual_sync": "complete_source_bound",
    "formal_Lean": "NOT_RUN",
    "reader_review": "142_of_142_source_bound",
    "wording_review": "original_Lead_decisions_resolved",
    "glossary_reader_paraphrases": [],
    "independent_AI_concurrence": "SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW",
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    "scope": "Owner-adopted reader revision. Lead and bilingual review concern the recorded text and premises; no new universal or Lean proof, canonical promotion or historical-label rewrite."
  }
}