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5b:T582,Найдём производную по правилу произведения, учитывая $ \bigl(e^{25-x}\bigr)' = -e^{25-x} $: \[ y' = (4x - 38)e^{25-x} + (2x^2 - 38x + 38)\cdot(-e^{25-x}) \] \[ = e^{25-x}\bigl[(4x - 38) - (2x^2 - 38x + 38)\bigr] \] \[ = e^{25-x}\bigl[-2x^2 + 42x - 76\bigr] \] \[ = -2\,e^{25-x}\bigl(x^2 - 21x + 38\bigr). \] Так как $ e^{25-x} > 0 $ при всех $ x $, знак $ y' $ противоположен знаку трёхчлена $ x^2 - 21x + 38 $. Корни квадратного уравнения $ x^2 - 21x + 38 = 0 $: \[ x = \frac{21 \pm \sqrt{441 - 152}}{2} = \frac{21 \pm \sqrt{289}}{2} = \frac{21 \pm 17}{2}. \] Значит $ x_1 = 2 $, $ x_2 = 19 $. Тогда $ y' = -2\,e^{25-x}(x - 2)(x - 19) $. Анализ знаков на числовой прямой: 1. при $ x < 2 $: $ (x-2)<0 $, $ (x-19)<0 $, произведение положительно, $ y' < 0 $ — функция убывает; 2. при $ 2 < x < 19 $: $ (x-2)>0 $, $ (x-19)<0 $, произведение отрицательно, $ y' > 0 $ — функция возрастает; 3. при $ x > 19 $: $ y' < 0 $ — функция убывает. В точке $ x = 2 $ производная меняет знак с «$ - $» на «$ + $» — это точка минимума. Ответ: $ 2 $.41:[["$","script",null,{"id":"collection-schema","type":"application/ld+json","dangerouslySetInnerHTML":{"__html":"{\"@context\":\"https://schema.org\",\"@type\":\"Course\",\"name\":\"Задачи по теме «Исследование произведений» — ЕГЭ профильный\",\"description\":\"Подборка задач ЕГЭ профильный уровень по теме «Исследование произведений». Включает 5 задач разной сложности. 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Включает 5 задач разной сложности. Подходит для целенаправленной отработки конкретного навыка перед экзаменом.","hidden_problem_count":0,"solved_count":0,"topic_id":"55baa8eb-7492-4c6b-9a97-f5cac87ca841","total_problem_count":5,"visible_problem_count":5},"created_at":"2026-08-31T10:38:35.80469Z","updated_at":"2026-08-31T16:23:22.775864Z","problem_count":5,"visible_count":5,"total_count":5,"pdf_unlocked":false},"serverCollectionProblems":{"items":[{"problem_id":"75742c9e-92b4-4693-9668-3952e7a546b3","collection_id":"74f274de-fb06-4086-8e65-c09d751918ac","display_order":1,"metadata":{},"created_at":"2026-08-31T10:38:35.80469Z","updated_at":"2026-08-31T10:38:35.80469Z","problem":{"id":"75742c9e-92b4-4693-9668-3952e7a546b3","short_id":16689,"task_number":9,"topic_id":"55baa8eb-7492-4c6b-9a97-f5cac87ca841","exam_id":"23f00ba7-bd19-4ca8-9349-5134d88397cf","difficulty":4,"content_text":"Найдите точку минимума функции \$ y = (7 - x) \\cdot e^{7-x} \$.","answer_text":"8","answer":"8","solution_text":"Найдём производную функции\n\$ y = (7 - x) e^{7-x} \$:\n\n\\[\ny' = -1 \\cdot e^{7-x} + (7 - x) \\cdot e^{7-x} \\cdot (-1) = -e^{7-x}(1 + 7 - x) = -e^{7-x}(8 - x).\n\\]\n\nНайдём критические точки:\n\n\\[\ny' = 0 \\Leftrightarrow\n8 - x = 0 \\Leftrightarrow\nx = 8.\n\\]\n\nОпределим знаки производной на промежутках:\n- При \$ x < 8 \$: \$ 8 - x > 0 \$, поэтому \$ y' = -e^{7-x} \\cdot (8 - x) < 0 \$ — функция убывает.\n- При \$ x > 8 \$: \$ 8 - x < 0 \$, поэтому \$ y' = -e^{7-x} \\cdot (8 - x) > 0 \$ — функция возрастает.\n\nЗначит, \$ x = 8 \$ — точка минимума.\n\nОтвет: 8.","explanation":"","score":1,"part":"1","metadata":{"answer_source":"llm-generated","pdf_url":"https://alexlarin.net/ege/2025/trvar505.pdf","solution_confidence":"high","solution_source":"llm-generated","source":"larin-trvar","suffix":"5","task_number_hint":12,"task_position":12,"tools_applied":{"classify":"2026-05-11T08:41:12Z","fill_solutions":"2026-05-11T08:15:26Z","format_content":"2026-05-17T07:04:27Z","multiversion_split":"2026-05-10T12:02:25Z"},"variant_label":"А. 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Проверьте, чтобы каждый ответ был\nзаписан в строке с номером соответствующего задания\nЧасть 2\nДля записи решений и ответов на задания 13–19 используйте БЛАНК ОТВЕТОВ №2.\nЗапишите сначала номер выполняемого задания (13, 14 и т.д.), а затем полное\nобоснованное решение и ответ. Ответы записывайте чётко и разборчиво.","pdf_url":"https://alexlarin.net/ege/2026/trvar519.pdf","position_in_pdf":12,"solution_confidence":"high","solution_source":"llm-generated","source":"larin-trvar","task_number_hint":12,"tools_applied":{"classify":"2026-05-07T19:44:37Z","fill_solutions":"2026-05-15T23:57:10Z","format_content":"2026-05-16T16:33:32Z","subagent_extract":"2026-05-15T20:39:05Z"},"variant_num":519,"year":2026},"created_at":"2026-05-23T14:02:11.48905Z","updated_at":"2026-08-30T22:39:08.368068Z","is_favorite":false,"is_solved":false,"is_reported":false,"topic":{"id":"55baa8eb-7492-4c6b-9a97-f5cac87ca841","name":"Исследование произведений","description":null},"tags":[{"id":"1007cfa3-afc8-42b1-b484-e851bfc08485","code":"tag-12","name":"Монотонность функции. Промежутки возрастания и убывания"},{"id":"14161c1f-2aff-4a19-a174-b71b0e3c1140","code":"tag-16","name":"Применение производной к исследованию функций и построению графиков"},{"id":"47c85d66-1ef5-4425-98bb-a75069df7dc2","code":"tag-68","name":"Показательная функция, её график"},{"id":"a04537df-8d08-476c-92fd-384e73a3e38a","code":"tag-143","name":"Точки экстремума функции"}]}},{"problem_id":"78d2ecb7-6eb2-46bc-b804-1e6b6eb61bd1","collection_id":"74f274de-fb06-4086-8e65-c09d751918ac","display_order":3,"metadata":{},"created_at":"2026-08-31T10:38:35.80469Z","updated_at":"2026-08-31T10:38:35.80469Z","problem":{"id":"78d2ecb7-6eb2-46bc-b804-1e6b6eb61bd1","short_id":16984,"task_number":9,"topic_id":"55baa8eb-7492-4c6b-9a97-f5cac87ca841","exam_id":"23f00ba7-bd19-4ca8-9349-5134d88397cf","difficulty":4,"content_text":"Найдите точку максимума функции \$ f(x) = \\sqrt{x}\\cdot(8 - \\sqrt{x}) + \\sqrt{2} \$.","answer_text":"16","answer":"16","solution_text":"Раскроем скобки: \$ f(x) = 8\\sqrt{x} - x + \\sqrt{2} \$. Область определения: \$ x \\geqslant 0 \$.\n\nПроизводная при \$ x > 0 \$:\n\n\\[\nf'(x) = \\frac{8}{2\\sqrt{x}} - 1 = \\frac{4}{\\sqrt{x}} - 1 = \\frac{4 - \\sqrt{x}}{\\sqrt{x}}.\n\\]\n\nПриравняем производную к нулю:\n\n\\[\nf'(x) = 0 \\Leftrightarrow \\sqrt{x} = 4 \\Leftrightarrow\nx = 16.\n\\]\n\nПри \$ 0 < x < 16 \$ имеем \$ \\sqrt{x} < 4 \$, поэтому \$ f'(x) > 0 \$ — функция возрастает; при \$ x > 16 \$ имеем \$ f'(x) < 0 \$ — функция убывает. Следовательно, \$ x = 16 \$ — точка максимума.\n\nОтвет: 16","explanation":"","score":1,"part":"1","metadata":{"condition_confidence":"high","condition_source":"larin-pdf","external_id":"larin-trvar-2026-V520-T12","is_legacy":false,"original_content_text":"Найдите точку максимума функции\n2\n)\n8(\n)\n(\n\n\n\n\nх\nх\nx\nf\n.\nНе забудьте перенести все ответы в бланк ответов № 1 в соответствии с\nинструкцией по выполнению работы.\nПроверьте, чтобы каждый ответ был записан в строке с номером\nсоответствующего задания\nЧасть 2\nДля записи решений и ответов на задания 13–19 используйте БЛАНК ОТВЕТОВ №2.\nЗапишите сначала номер выполняемого задания (13, 14 и т.д.), а затем полное\nобоснованное решение и ответ. Ответы записывайте чётко и разборчиво.","pdf_url":"https://alexlarin.net/ege/2026/trvar520.pdf","position_in_pdf":12,"solution_confidence":"high","solution_source":"larin-forum","source":"larin-trvar","source_post_indices":"[14]","task_number_hint":12,"tools_applied":{"classify":"2026-05-07T19:52:20Z","format_content":"2026-05-16T18:49:08Z","subagent_extract":"2026-05-15T20:39:06Z"},"variant_num":520,"year":2026},"created_at":"2026-05-23T14:02:31.501618Z","updated_at":"2026-08-30T22:39:08.368068Z","is_favorite":false,"is_solved":false,"is_reported":false,"topic":{"id":"55baa8eb-7492-4c6b-9a97-f5cac87ca841","name":"Исследование произведений","description":null},"tags":[{"id":"1007cfa3-afc8-42b1-b484-e851bfc08485","code":"tag-12","name":"Монотонность функции. Промежутки возрастания и убывания"},{"id":"14161c1f-2aff-4a19-a174-b71b0e3c1140","code":"tag-16","name":"Применение производной к исследованию функций и построению графиков"},{"id":"a04537df-8d08-476c-92fd-384e73a3e38a","code":"tag-143","name":"Точки экстремума функции"},{"id":"da67531d-0011-489f-adab-84744c7aa977","code":"tag-208","name":"Преобразования выражений, включающих корни натуральной степени"}]}},{"problem_id":"d516f47a-9872-4317-95d8-119c6aec902f","collection_id":"74f274de-fb06-4086-8e65-c09d751918ac","display_order":4,"metadata":{},"created_at":"2026-08-31T10:38:35.80469Z","updated_at":"2026-08-31T10:38:35.80469Z","problem":{"id":"d516f47a-9872-4317-95d8-119c6aec902f","short_id":18461,"task_number":9,"topic_id":"55baa8eb-7492-4c6b-9a97-f5cac87ca841","exam_id":"23f00ba7-bd19-4ca8-9349-5134d88397cf","difficulty":4,"content_text":"Найдите точку максимума функции \$ y = (x + 9) \\cdot e^{12-x} \$.","answer_text":"\$ -8 \$","answer":"\$ -8 \$","solution_text":"Функция определена при всех \$ x \\in \\mathbb{R} \$. Исследуем функцию и найдём её промежутки возрастания и убывания, для этого найдём её производную:\n\\[ y' = 1 \\cdot e^{12-x} + (x + 9) \\cdot e^{12-x} \\cdot (-1) = e^{12-x}(1 - (x + 9)) = e^{12-x}(-8 - x) = -e^{12-x}(8 + x). \\]\n\nНайдём нули производной:\n\\[ y' = 0 \\quad \\Rightarrow \\quad x = -8. \\]\n\nНули производной и точки, в которых она не существует, разбивают область определения производной на промежутки, на каждом из которых она непрерывна и принимает значения одного знака. Найдём знаки производной на каждом из таких промежутков.\n\nПри \$ x \\in (-\\infty; -8) \$ производная положительна, то есть функция \$ y = y(x) \$ возрастает; при \$ x \\in (-8; +\\infty) \$ производная отрицательна, то есть функция убывает. Следовательно, \$ x = -8 \$ является точкой максимума.","explanation":null,"score":1,"part":"1","metadata":{"confidence":"high","content_hash":"260eaca99d8eb791d0a66944","hold_reasons":[],"ingest_batch":"resit2026-v1","region":"Сибирь","reserve_key":"resit_2026|12.3:Сибирь","solution_source":"shkolkovo","source":"shkolkovo","task_number":12,"variant":3,"wave":"resit_2026","year":2026},"created_at":"2026-07-10T06:04:04.537731Z","updated_at":"2026-08-31T14:47:37.91245Z","is_favorite":false,"is_solved":false,"is_reported":false,"solution_images":[{"id":"deecfaea-75a0-4967-9714-e72534d9e5da","url":"https://db.sdamex.com/storage/v1/object/public/math_prof_imgs/svg/resit2026_12_3_s_934c1e8a.svg","type":"svg","description":"","display_order":0}],"topic":{"id":"55baa8eb-7492-4c6b-9a97-f5cac87ca841","name":"Исследование произведений","description":null},"tags":[{"id":"1007cfa3-afc8-42b1-b484-e851bfc08485","code":"tag-12","name":"Монотонность функции. Промежутки возрастания и убывания"},{"id":"36f3cc51-f003-4ed2-927d-039e1428bef9","code":"tag-49","name":"Наибольшее и наименьшее значения функции"},{"id":"a04537df-8d08-476c-92fd-384e73a3e38a","code":"tag-143","name":"Точки экстремума функции"}]}},{"problem_id":"ba8f2281-2819-4a90-8740-c039011fddd0","collection_id":"74f274de-fb06-4086-8e65-c09d751918ac","display_order":5,"metadata":{},"created_at":"2026-08-31T10:38:35.80469Z","updated_at":"2026-08-31T10:38:35.80469Z","problem":{"id":"ba8f2281-2819-4a90-8740-c039011fddd0","short_id":17213,"task_number":9,"topic_id":"55baa8eb-7492-4c6b-9a97-f5cac87ca841","exam_id":"23f00ba7-bd19-4ca8-9349-5134d88397cf","difficulty":5,"content_text":"Найдите наименьшее значение функции \$ y = \\dfrac{x^3 + 2x^2 + 72}{x} \$ на отрезке \$ [1; 6] \$.","answer_text":"39","answer":"39","solution_text":"Функция: \$ y = \\dfrac{x^3 + 2x^2 + 72}{x} = x^2 + 2x + \\dfrac{72}{x} \$ на \$ [1; 6] \$.\n\nПроизводная:\n\n\\[\ny' = 2x + 2 - \\frac{72}{x^2} = \\frac{2x^3 + 2x^2 - 72}{x^2} = \\frac{2(x^3 + x^2 - 36)}{x^2}.\n\\]\n\nИщем \$ y' = 0 \$: \$ x^3 + x^2 - 36 = 0 \$. Подбором: \$ x = 3 \$ — корень (\$ 27 + 9 - 36 = 0 \$). Разложим:\n\n\\[\nx^3 + x^2 - 36 = (x - 3)(x^2 + 4x + 12).\n\\]\n\nКвадратное уравнение \$ x^2 + 4x + 12 = 0 \$ имеет \$ D = 16 - 48 < 0 \$ — действительных корней нет.\n\nЕдинственная критическая точка \$ x = 3 \\in [1; 6] \$.\n\nПри \$ x < 3 \$: \$ y' < 0 \$ (функция убывает); при \$ x > 3 \$: \$ y' > 0 \$ (функция возрастает). Значит, \$ x = 3 \$ — точка минимума.\n\n\\[\ny(3) = \\frac{27 + 18 + 72}{3} = \\frac{117}{3} = 39.\n\\]\n\nДополнительно: \$ y(1) = 75 \$, \$ y(6) = 60 \$ — минимум действительно в точке \$ x = 3 \$.\n\nОтвет: 39.","explanation":"","score":1,"part":"1","metadata":{"external_id":"larin-trvar-2026-V531-T12","is_legacy":false,"pdf_url":"https://alexlarin.net/ege/2026/trvar531.pdf","position_in_pdf":12,"pre_layout_content":"Найдите наименьшее значение функции\nx\nx\nx\nу\n72\n2\n2\n3\n\n\n\nна отрезке \n\n6;1\n.\nНе забудьте перенести все ответы в бланк ответов № 1 в соответствии с\nинструкцией по выполнению работы. Проверьте, чтобы каждый ответ был\nзаписан в строке с номером соответствующего задания\nЧасть 2\nДля записи решений и ответов на задания 13–19 используйте БЛАНК ОТВЕТОВ №2.\nЗапишите сначала номер выполняемого задания (13, 14 и т.д.), а затем полное\nобоснованное решение и ответ. Ответы записывайте чётко и разборчиво.","solution_confidence":"high","solution_source":"llm-generated","source":"larin-trvar","task_number_hint":12,"tools_applied":{"classify":"2026-05-07T19:57:26Z","fill_solutions":"2026-05-16T05:49:39Z","fix_layout":"2026-05-16 08:15:14","format_content":"2026-05-17T06:48:53Z"},"variant_num":531,"year":2026},"created_at":"2026-05-23T14:04:39.442686Z","updated_at":"2026-08-30T22:39:08.368068Z","is_favorite":false,"is_solved":false,"is_reported":false,"topic":{"id":"55baa8eb-7492-4c6b-9a97-f5cac87ca841","name":"Исследование произведений","description":null},"tags":[{"id":"14161c1f-2aff-4a19-a174-b71b0e3c1140","code":"tag-16","name":"Применение производной к исследованию функций и построению графиков"},{"id":"36f3cc51-f003-4ed2-927d-039e1428bef9","code":"tag-49","name":"Наибольшее и наименьшее значения функции"},{"id":"a04537df-8d08-476c-92fd-384e73a3e38a","code":"tag-143","name":"Точки экстремума функции"}]}}],"total":5},"serverExams":[{"id":"a217734d-d4f5-4d66-ba1a-5248bb381f09","code":"dvi-math","name":"ДВИ МГУ (математика)","subject_id":"d1be5154-7be0-4c8c-92d3-c486ea0a1a57","subject_name":"Математика","subject_code":"math","program_id":"1aed6121-06cc-4a5a-87f5-77de484ebb36","program_code":"dvi","exam_type":"ДВИ","description":"Дополнительное вступительное испытание МГУ им. М.В. Ломоносова по математике. С 2026 года экзамен состоит из двух частей: лёгкий тест с кратким ответом и традиционная письменная часть с полным решением.","has_part2":true,"max_task_number":20,"exam_date":"2026-07-11","metadata":{"duration_minutes":180,"max_secondary_score":100,"mock_exam_enabled":true,"part2_tasks":6,"parts":[{"answer_format":"detailed","part":1,"scale":"linear","slots_in_variant":10,"tasks":[1,2,3,4,5,6,7,8,9,10,11,12,13],"weight":30},{"answer_format":"detailed","part":2,"scale":"score_conversion","slots_in_variant":6,"tasks":[14,15,16,17,18,19,20],"weight":70}],"score_conversion":[2,30,30,35,35,35,35,35,40,40,40,40,40,45,45,45,45,45,50,50,50,50,50,55,55,55,55,55,60,60,60,60,60,65,65,65,65,65,70,70,70,70,70,75,75,75,75,75,80,80,80,80,80,85,85,85,85,85,90,90,90,90,90,95,95,95,95,95,100,100,100],"scoring_table":{"1":1,"2":1,"3":1,"4":1,"5":1,"6":1,"7":1,"8":1,"9":1,"10":1,"11":1,"12":1,"13":1,"14":10,"15":10,"16":10,"17":10,"18":10,"19":10,"20":10},"structure_2026":{"consultation_video_url":"https://disk.360.yandex.ru/d/NbP139HUZpCPDA","exam_2026":{"exam_date":"2026-07-11","note":"Реальный основной день; в образце Ш12622 тест был из 15 задач.","test_task_count":10,"written_task_count":6},"exam_portal":"https://exam.msu.ru/","official_exam_page":"https://cpk.msu.ru/exam/101","parts":["base","written"],"test_task_count":15},"task_count":7,"task_numbering":"part_relative","year":2026},"created_at":"2026-06-10T20:49:34.08312Z","updated_at":"2026-09-01T00:33:23.788959Z","problem_count":1008,"theory_count":36,"system_collection_count":190},{"id":"3a8471c5-2a11-4932-ba3e-7ab7d1ae7a1d","code":"ege-base","name":"Математика (база) ЕГЭ","subject_id":"d1be5154-7be0-4c8c-92d3-c486ea0a1a57","subject_name":"Математика","subject_code":"math","old_code":"math_base","program_id":"175d2aa1-bdcb-4df2-9710-7a61057ddd66","program_code":"ege","exam_type":"ЕГЭ","description":"Базовый ЕГЭ по математике для получения аттестата","has_part2":false,"max_task_number":21,"exam_date":"2026-06-04","metadata":{"duration_minutes":180,"max_score":21,"max_secondary_score":5,"mock_exam_enabled":true,"part1_tasks":21,"part2_tasks":0,"score_conversion":[2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5],"scoring_table":{"1":1,"2":1,"3":1,"4":1,"5":1,"6":1,"7":1,"8":1,"9":1,"10":1,"11":1,"12":1,"13":1,"14":1,"15":1,"16":1,"17":1,"18":1,"19":1,"20":1,"21":1},"year":2026},"created_at":"2025-11-09T12:48:12.440483Z","updated_at":"2025-11-09T12:48:12.440483Z","problem_count":8561,"theory_count":85,"system_collection_count":111},{"id":"23f00ba7-bd19-4ca8-9349-5134d88397cf","code":"ege-prof","name":"Математика (профиль) ЕГЭ","subject_id":"d1be5154-7be0-4c8c-92d3-c486ea0a1a57","subject_name":"Математика","subject_code":"math","old_code":"math_prof","program_id":"175d2aa1-bdcb-4df2-9710-7a61057ddd66","program_code":"ege","exam_type":"ЕГЭ","description":"Профильный ЕГЭ по математике для поступающих в технические вузы","has_part2":true,"max_task_number":20,"exam_date":"2026-06-08","metadata":{"duration_minutes":235,"max_score":33,"max_secondary_score":100,"max_task_number":20,"mock_exam_enabled":true,"part1_tasks":13,"part2_tasks":7,"score_conversion":[0,6,11,17,22,27,34,40,46,52,58,64,70,72,74,76,78,80,82,84,86,88,90,92,94,95,96,97,98,99,100,100,100,100],"scoring_table":{"1":1,"2":1,"3":1,"4":1,"5":1,"6":1,"7":1,"8":1,"9":1,"10":1,"11":1,"12":1,"13":1,"14":2,"15":3,"16":2,"17":2,"18":3,"19":4,"20":4},"year":2026},"created_at":"2025-11-09T12:48:12.440483Z","updated_at":"2026-08-30T23:51:20.728833Z","problem_count":2491,"theory_count":123,"system_collection_count":255},{"id":"b0db9506-ddcf-4200-8815-7a709ea0dc4e","code":"oge-math","name":"ОГЭ (математика)","subject_id":"d1be5154-7be0-4c8c-92d3-c486ea0a1a57","subject_name":"Математика","subject_code":"math","program_id":"ff8f63b8-bfcc-47f3-9d55-bc666ee30237","program_code":"oge","exam_type":"ОГЭ","description":"Основной государственный экзамен по математике (9 класс): 25 заданий, 235 минут. 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