{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261005-advance-20261006-r02-quadratic-function-0066", "pilot_number": "R0068-34566", "number": 34566, "family_id": "qf-quadratic-function-extremum", "title": { "zh-Hant": "第 34566 題：全實數二次函數的全域極值", "en": "Question 34566: Global Extremum over All Real Inputs" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261005-advance-20261006-r02-quadratic-function-0066", "batch_id": "batch-20261005-advance-20261006-r02-quadratic-function-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-quadratic-function" ], "title_zh": "全實數二次函數的全域極值", "stem_zh": "在全實數論域，指定 q:R→R，q(x)=-4(x−(8))²+(3)，a=-4≠0。指出全域最小值或全域最大值，並給出該值；不限制成有限輸入區間。", "answer": { "kind": "short_text", "value": "全域最大值=3", "explanation_zh": "全域最大值=3。平方項非負，a>0 時 q(x)≥k，a<0 時 q(x)≤k，並在 x=h 取得等號。" }, "difficulty": { "band": "upper_secondary", "level": 2, "skills": [ "全實數二次函數的全域極值" ] }, "variant": { "family_id": "qf-quadratic-function-extremum", "seed": 202610062066, "parameters": { "a": -4, "h": 8, "k": 3, "domain": "reals", "operation": "quadratic_extremum", "semantic_template": "qf-quadratic-function-extremum/{\"inputs\":[\"maximum\",\"3\"],\"operation\":\"quadratic_extremum\"}" } }, "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": "2026-10-05", "contract_version": "ocme-question-factory-v0.1" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20261005-advance-20261006-r02-quadratic-function-0066", "batch_id": "batch-20261005-advance-20261006-r02-quadratic-function-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "12e1205a40127c0ff2caf57a334f3beac8cd878170b1ba815507bde18b65957d", "family_id": "qf-quadratic-function-extremum", "seed": 202610062066, "semantic_template": "qf-quadratic-function-extremum/{\"inputs\":[\"maximum\",\"3\"],\"operation\":\"quadratic_extremum\"}" }, "fields": { "title": "Global Extremum over All Real Inputs", "stem": "On the full real domain, let q:R→R be q(x)=(-4)(x−(8))²+(3), with a≠0. Give the global minimum or global maximum value, according to the coefficient sign. Do not substitute a restricted interval.", "answer": { "kind": "short_text", "value": "global maximum=3", "explanation": "Because (x−h)²≥0, a>0 gives a global minimum and a<0 a global maximum, attained at x=h. Here global maximum=3; the function is unbounded in the other direction." }, "skills": [ "Global Extremum over All Real Inputs" ], "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": "在全實數論域，指定 q:R→R，q(x)=-4(x−(8))²+(3)，a=-4≠0。指出全域最小值或全域最大值，並給出該值；不限制成有限輸入區間。", "en": "On the full real domain, let q:R→R be q(x)=(-4)(x−(8))²+(3), with a≠0. Give the global minimum or global maximum value, according to the coefficient sign. Do not substitute a restricted interval." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source roles = [qf-quadratic-function-extremum, [a, domain, h, k, operation]]", "after": "derived = [quadratic_extremum, reals]", "relation": "implies", "rule_id": "R68_BANK_specific_keys_operations_and_numeric_guards", "conditions": [], "explanation": { "zh-Hant": "按來源分bank：legacy raw SAFE/衍生BigInt無界，BASE Number.isInteger不偷加SAFE/百萬界。underscore operation、domain及寬/嚴keys各依本bank；原JSON完整保存。", "en": "Separate source banks:legacy RAW SAFE with unbounded derived BigInts versus BASE Number.isInteger without invented SAFE/million caps.Operations,domains and permissive/exact keys follow THIS bank.Keep ORIGINAL JSON intact." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反向核對本題的原始已知量、論域與本步條件；保留原題/英文/MKO/AST/狀態，不把單向推論倒成等價，也不把有限核對當獨立審查或一般形式證明。", "en": "Reverse-check THIS question’s ORIGINAL givens,domain and step premises.Preserve originals/English/MKOs/ASTs/status;do not reverse a one-way implication or call finite checks independent review or general formal proof." } }, "arithmetic": null }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "source roles = [-4, 8, 3]", "after": "derived = [q(x)=a*(x-h)^2+k, x in R]", "relation": "implies", "rule_id": "Quadratic_FULL_real_vertex_form_nonzero_leading_coefficient", "conditions": [], "explanation": { "zh-Hant": "先驗a非0與完整實數論域，原函數是頂點式而非a*x²+h*x+k。所有下列變形都保持原h/k的角色；不偷換有限區間問題。", "en": "FIRST qualify nonzero a and the FULL real domain.The ORIGINAL function is vertex form,not a*x²+h*x+k.Keep h/k roles unchanged and do not silently replace the domain by a bounded interval." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反向核對本題的原始已知量、論域與本步條件；保留原題/英文/MKO/AST/狀態，不把單向推論倒成等價，也不把有限核對當獨立審查或一般形式證明。", "en": "Reverse-check THIS question’s ORIGINAL givens,domain and step premises.Preserve originals/English/MKOs/ASTs/status;do not reverse a one-way implication or call finite checks independent review or general formal proof." } }, "arithmetic": null }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "Quadratic ORIGINAL x minus h := sub(8, 8)", "after": "Quadratic ORIGINAL x minus h = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按原操作數/次序精確運算；乘/減/除明示兩操作數，legacy BigInt衍生量不重套raw SAFE界，BASE Number來源的型別/表示條件另驗，不以浮點近似湊答案。", "en": "Calculate exactly from ORIGINAL ordered operands;mul/sub/div have EXPLICITLY TWO operands.Legacy derived BigInts are not recapped at RAW SAFE;check BASE Number type/representation conditions separately.Do not use floating approximations to force an answer." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Quadratic ORIGINAL x minus h，不以小數近似替代。", "en": "Recompute Quadratic ORIGINAL x minus h with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "8", "8" ], "output": "0", "label": "Quadratic ORIGINAL x minus h" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "Quadratic ORIGINAL displacement squared := mul(0, 0)", "after": "Quadratic ORIGINAL displacement squared = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按原操作數/次序精確運算；乘/減/除明示兩操作數，legacy BigInt衍生量不重套raw SAFE界，BASE Number來源的型別/表示條件另驗，不以浮點近似湊答案。", "en": "Calculate exactly from ORIGINAL ordered operands;mul/sub/div have EXPLICITLY TWO operands.Legacy derived BigInts are not recapped at RAW SAFE;check BASE Number type/representation conditions separately.Do not use floating approximations to force an answer." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Quadratic ORIGINAL displacement squared，不以小數近似替代。", "en": "Recompute Quadratic ORIGINAL displacement squared with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "0", "0" ], "output": "0", "label": "Quadratic ORIGINAL displacement squared" } }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "Quadratic ORIGINAL a times displacement squared := mul(-4, 0)", "after": "Quadratic ORIGINAL a times displacement squared = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按原操作數/次序精確運算；乘/減/除明示兩操作數，legacy BigInt衍生量不重套raw SAFE界，BASE Number來源的型別/表示條件另驗，不以浮點近似湊答案。", "en": "Calculate exactly from ORIGINAL ordered operands;mul/sub/div have EXPLICITLY TWO operands.Legacy derived BigInts are not recapped at RAW SAFE;check BASE Number type/representation conditions separately.Do not use floating approximations to force an answer." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Quadratic ORIGINAL a times displacement squared，不以小數近似替代。", "en": "Recompute Quadratic ORIGINAL a times displacement squared with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-4", "0" ], "output": "0", "label": "Quadratic ORIGINAL a times displacement squared" } }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "Quadratic ORIGINAL weighted square plus k := sum(0, 3)", "after": "Quadratic ORIGINAL weighted square plus k = 3", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按原操作數/次序精確運算；乘/減/除明示兩操作數，legacy BigInt衍生量不重套raw SAFE界，BASE Number來源的型別/表示條件另驗，不以浮點近似湊答案。", "en": "Calculate exactly from ORIGINAL ordered operands;mul/sub/div have EXPLICITLY TWO operands.Legacy derived BigInts are not recapped at RAW SAFE;check BASE Number type/representation conditions separately.Do not use floating approximations to force an answer." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Quadratic ORIGINAL weighted square plus k，不以小數近似替代。", "en": "Recompute Quadratic ORIGINAL weighted square plus k with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "0", "3" ], "output": "3", "label": "Quadratic ORIGINAL weighted square plus k" } }, { "id": "S007", "uses": [ "G001", "S006" ], "before": "source roles = [-4, 8, 3]", "after": "derived = [(x-h)^2>=0 for ALL real x, q(x)<=k, q(h)=k, 全域最大值=3]", "relation": "implies", "rule_id": "Quadratic_UNIVERSAL_square_sign_global_bound_AND_attainment", "conditions": [], "explanation": { "zh-Hant": "對每個實數x平方非負，乘正a保≥、負a反向成≤；再加k得全域界，且x=h真的達到k。另一方向隨|x-h|→∞無界，所以不假稱兩方向都有有限全域最值。", "en": "For EVERY real x,the square is nonnegative.Positive a preserves≥;negative a reverses it to≤.Adding k gives the GLOBAL bound,and x=h ATTAINS k.The other direction is unbounded as|x-h|→infinity;do not invent finite extrema in both directions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反向核對本題的原始已知量、論域與本步條件；保留原題/英文/MKO/AST/狀態，不把單向推論倒成等價，也不把有限核對當獨立審查或一般形式證明。", "en": "Reverse-check THIS question’s ORIGINAL givens,domain and step premises.Preserve originals/English/MKOs/ASTs/status;do not reverse a one-way implication or call finite checks independent review or general formal proof." } }, "arithmetic": null } ], "math_model": { "result": "全域最大值=3" }, "conclusion": { "canonical_value": "全域最大值=3", "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." }, "source": { "input_commit": "0f7be61e2c22317a4075561dc822c87d881a50db", "question_binding": { "batch_id": "batch-20261005-advance-20261006-r02-quadratic-function-foundations", "batch_path": "public/data/questions/batches/batch-20261005-advance-20261006-r02-quadratic-function-foundations/questions.jsonl", "batch_sha256": "32ddadfe8c46e9f3d07fce31517b78bd2cd661610eee7e0d814f91fecbc51be4", "original_line": 66, "original_line_sha256": "9f399f012fee3f4ddf6ce7053ed917a8d7eac71a85feb323885fe4c013955874", "question_compact_sha256": "12e1205a40127c0ff2caf57a334f3beac8cd878170b1ba815507bde18b65957d", "english_row_compact_sha256": "4a7ecbc63f4db91784e53688858770b46c0e2c3fb678d204d8b9d88aea303df5", "english_manifest_path": "public/data/i18n/editions/batch-20261005-advance-20261006-r02-quadratic-function-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261005-advance-20261006-r02-quadratic-function-foundations.en.jsonl" }, "mko_bindings": { "mko-quadratic-function": { "sha256": "439528a9247668c73092d919fb031abee74709f0633c254061b2256ed99f6441", "english_sha256": "302ee0311ea4ba78b81f3c48951bf1dc495a2b158c4c0e46b3f8b3b0ff72bf67", "original_path": "public/data/mko/quadratic-function.json", "english_path": "public/data/i18n/editions/mko-quadratic-function.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "345142be27643acca9d0fec2422ac7311b85bf142709391cd6b5b3a33c03cf8e", "original_candidate_sha256": "ccdad953a5eb5f627e0ea9c511b50e25f00ee97f6fda8bd9980d3ecfd7d51d24", "original_candidate_line_sha256": "ccdad953a5eb5f627e0ea9c511b50e25f00ee97f6fda8bd9980d3ecfd7d51d24" }, "historical_source_state": { "review_state": { "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-20261026-012-p69", "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": { "semantic_review_sha256": "e7a54104fa02c235f7591329bd6db8d626dd74bf0fae29c557b1f269c85066ed", "question_first_witness_sha256": "8401a3c2064c1275a6a7ea8f00412fa12286cec7b2db6c305ff9dba3724ae229" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 7, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "b2b618257731ce7baa0d27d477203fa721b7426512c8590d612104a0737d3123", "arithmetic_steps": 4 }, "uses_answer_as_premise": false, "human_review": "NOT_PERFORMED", "formalization": "not_formalized", "Lean": "NOT_RUN", "Canonical_truth_promotion": false, "authority": "Owner-authorized reviewed reader publication; original question/candidate states remain historical and unchanged." } }