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Консультация № 177103
07.03.2010, 00:16
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здавствуйте,помогите пожалуйста решить задание
20. Для каждого A решите неравенство log^2 основание2 (x)-A*log основание 2 (x) -2*A^2 <0
20. Для каждого A решите неравенство log^2 основание2 (x)-A*log основание 2 (x) -2*A^2 <0
Обсуждение
Неизвестный
07.03.2010, 09:14
общий
09.03.2010, 23:10
это ответ
Здравствуйте, Артём Бортников.
Делаем замену log2(x)=y. (log2(x) обозначает логарифм x по основанию 2)
Получаем квадратное неравенство y2-A*y-2*A2<0.
Решаем уравнение y2-A*y-2*A2=0. Оно имеет два корня: y1=-A, y2=2*A.
Теперь решаем неравенство (методом интервалов).
При A>0 решение -A<y<2*A
При A=0 уравнение принимает вид y2<0 - корней нет
При A<0 решение 2*A<y<-A
Теперь подставляем суюда y=log2(x)
При A>0 -A<log2(x)<2*A => 2-A<x<4A
При A=0 корней нет
При A>0 2*A<log2(x)<-A => 4A<x<2-A
Делаем замену log2(x)=y. (log2(x) обозначает логарифм x по основанию 2)
Получаем квадратное неравенство y2-A*y-2*A2<0.
Решаем уравнение y2-A*y-2*A2=0. Оно имеет два корня: y1=-A, y2=2*A.
Теперь решаем неравенство (методом интервалов).
При A>0 решение -A<y<2*A
При A=0 уравнение принимает вид y2<0 - корней нет
При A<0 решение 2*A<y<-A
Теперь подставляем суюда y=log2(x)
При A>0 -A<log2(x)<2*A => 2-A<x<4A
При A=0 корней нет
При A>0 2*A<log2(x)<-A => 4A<x<2-A
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