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Консультация № 196630
10.10.2019, 10:28
0.00 руб.
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6
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Здравствуйте, уважаемые эксперты! Прошу вас ответить на следующий вопрос:
Докажите что a^2*b+a*b^2+1<a+b+a^2*b^2
Докажите что a^2*b+a*b^2+1<a+b+a^2*b^2
Обсуждение
10.10.2019, 14:42
общий
это ответ
Здравствуйте, wwesmack!
Требуется доказать неравенство:
a2b+ab2+1<a+b+a2b2 при a>1, b>1.
Собираем в правой части члены по степеням а, получим:
a2(b2-b)- a(b2-1) + b-1 >0, a2b(b-1) - a(b-1)(b+1) + b -1>0.
Так как b> 1, сокращаем на b-1:
a2b - a(b+1) + 1 > 0.
Собираем члены по степеням b:
(a2-a)b - a +1>0, a(a-1)b -(a-1)>0.
Так как a > 1, сокращаем на a-1:
ab -1 > 0.
В результате равносильных преобразований мы получили верное неравенство,
следовательно, исходное неравенство верно.
Требуется доказать неравенство:
a2b+ab2+1<a+b+a2b2 при a>1, b>1.
Собираем в правой части члены по степеням а, получим:
a2(b2-b)- a(b2-1) + b-1 >0, a2b(b-1) - a(b-1)(b+1) + b -1>0.
Так как b> 1, сокращаем на b-1:
a2b - a(b+1) + 1 > 0.
Собираем члены по степеням b:
(a2-a)b - a +1>0, a(a-1)b -(a-1)>0.
Так как a > 1, сокращаем на a-1:
ab -1 > 0.
В результате равносильных преобразований мы получили верное неравенство,
следовательно, исходное неравенство верно.
5
10.10.2019, 14:58
общий
Адресаты:
Не совсем понял строчки: "Так как b> 1, сокращаем на b-1" и "Так как a > 1, сокращаем на a-1". Объясните пожалуйста поподробней
10.10.2019, 15:05
общий
Адресаты:
Поздравляю! 
Об авторе:
"Если вы заметили, что вы на стороне большинства, —
это верный признак того, что пора меняться." Марк Твен
"Если вы заметили, что вы на стороне большинства, —
это верный признак того, что пора меняться." Марк Твен
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