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Консультация № 191260
03.08.2017, 01:56
0.00 руб.
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Здравствуйте, уважаемые эксперты! Прошу вас ответить на следующий вопрос:
Пусть х1,х2...хn - некоторые числа,принадлежащие отрезку [0;1]. Докажите, что на этом отрезке найдется такое число х,что (см.фото)
Пусть х1,х2...хn - некоторые числа,принадлежащие отрезку [0;1]. Докажите, что на этом отрезке найдется такое число х,что (см.фото)
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Обсуждение
03.08.2017, 08:23
общий
это ответ
Здравствуйте, vlaad!
Пусть
f(x) = (|x - x1| + |x - x2| + ... + |x - xn|)/n.
Найдим значения f(x) не концах интервала [0,1]:
f(0) = (x1 + x2 + ... + xn)/n,
f(1) = (1 - x1 + 1 - x2 + ... + 1 - xn1)/n = 1 - (x1 + x2 + ... + xn)/n = 1 - f(0)
(Мы смогли избавиться от модулей, так как 0 [$8804$] xk [$8804$] 1). Вспоминая свойства среднего арифметического, заключаем, что 0[$8804$]f(0)[$8804$]1, 0[$8804$]f(1)[$8804$]1.
Ясно, что либо f(0)[$8805$]1/2 и f(1)[$8804$]1/2, либо f(1)[$8805$]1/2 и f(0)[$8804$]1/2. Поскольку f(x) непрерывна, при изменении x от 0 до 1 она принимает все значения между f(0) и f(1), и в интервале [0,1] найдется х при котором f(x) = 1/2.
Пусть
f(x) = (|x - x1| + |x - x2| + ... + |x - xn|)/n.
Найдим значения f(x) не концах интервала [0,1]:
f(0) = (x1 + x2 + ... + xn)/n,
f(1) = (1 - x1 + 1 - x2 + ... + 1 - xn1)/n = 1 - (x1 + x2 + ... + xn)/n = 1 - f(0)
(Мы смогли избавиться от модулей, так как 0 [$8804$] xk [$8804$] 1). Вспоминая свойства среднего арифметического, заключаем, что 0[$8804$]f(0)[$8804$]1, 0[$8804$]f(1)[$8804$]1.
Ясно, что либо f(0)[$8805$]1/2 и f(1)[$8804$]1/2, либо f(1)[$8805$]1/2 и f(0)[$8804$]1/2. Поскольку f(x) непрерывна, при изменении x от 0 до 1 она принимает все значения между f(0) и f(1), и в интервале [0,1] найдется х при котором f(x) = 1/2.
5
Спасибо!
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