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Консультация № 196940
03.11.2019, 12:50
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Уважаемые эксперты! Пожалуйста, ответьте на вопрос:
Найдите все значения параметра а, при каждом из которых имеет ровно один отрицатель¬ный корень уравнение
х^4 + (а - 1)х^3 + х^2 + (а - 1)х + 1 = 0
Найдите все значения параметра а, при каждом из которых имеет ровно один отрицатель¬ный корень уравнение
х^4 + (а - 1)х^3 + х^2 + (а - 1)х + 1 = 0
Обсуждение
04.11.2019, 11:39
общий
Адресаты:
Я заменил a-1 на 2·b , стал махинировать , но запутался в сложных выкладках, ума не хватает.
Если математики не помогут Вам, то по Вашей просьбе я дам задание Маткаду. Он не покажет манипуляции с много-членами, но выдаст правильный Ответ, к которому Вам будет легче пробираться, видя конечную цель.
Если математики не помогут Вам, то по Вашей просьбе я дам задание Маткаду. Он не покажет манипуляции с много-членами, но выдаст правильный Ответ, к которому Вам будет легче пробираться, видя конечную цель.
19.08.2022, 20:19
общий
это ответ
Здравствуйте, kovalenina!
Свободный член заданного уравнения равен единице, поэтому если заданное уравнение имеет ровно один отрицательный корень, то кратность этого корня равна двум или четырём, и ноль не является корнем, то есть
Поэтому обе части заданного уравнения можно разделить на 
Тогда получим
Введём переменную
Тогда 
С новой переменной из уравнения (1) получим
Так как
-- единственный отрицательный корень заданного уравнения, то 
должно принимать своё значение только при этом значении 
Рассмотрим функцию 
при 
Приравняв к нулю её первую производную, получим
Левее этой точки рассматриваемая функция возрастает, правее -- убывает (рисунок в прикреплённом файле). Тогда
и из уравнения (2) получим
Ответ:
Свободный член заданного уравнения равен единице, поэтому если заданное уравнение имеет ровно один отрицательный корень, то кратность этого корня равна двум или четырём, и ноль не является корнем, то есть
Поэтому обе части заданного уравнения можно разделить на Тогда получимВведём переменную
Тогда С новой переменной из уравнения (1) получимТак как
-- единственный отрицательный корень заданного уравнения, то должно принимать своё значение только при этом значении Рассмотрим функцию при Приравняв к нулю её первую производную, получимЛевее этой точки рассматриваемая функция возрастает, правее -- убывает (рисунок в прикреплённом файле). Тогда
и из уравнения (2) получимОтвет:
Прикрепленные файлы:
Об авторе:
Facta loquuntur.
Facta loquuntur.
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