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Консультация № 200307
22.02.2021, 14:49
0.00 руб.
23.02.2021, 03:36
0
1
1
Здравствуйте! У меня возникли сложности с таким вопросом: Пусть функция y=f(x) дифференцируема на R . Верно ли, что
а) если f '(x)>0 для всех x , то f(x)>0 для всех x;
б) если f '(3)=0, то функция f имеет максимум или минимум в точке x=3.
а) если f '(x)>0 для всех x , то f(x)>0 для всех x;
б) если f '(3)=0, то функция f имеет максимум или минимум в точке x=3.
Обсуждение
23.02.2021, 03:47
общий
это ответ
Здравствуйте, Nekro!
а) Неверно. Например, для функции f(x) = arctg x её производная f '(x) = 1/1+x[sup]2[/sup] строго положительна при всех x [$8712$] R, но f(x) > 0 только при x > 0. Дополнительно должно выполняться условие
б) Неверно. Например, для функции f(x) = (x-3)[sup]3[/sup] её производная f '(x) = 3(x-3)[sup]2[/sup] равна нулю в точке x = 3, но функция f(x) не имеет в этой точке максимума или минимума (только перегиб). Дополнительно должно выполняться условие f "(3) [$8800$] 0.
а) Неверно. Например, для функции f(x) = arctg x её производная f '(x) = 1/1+x[sup]2[/sup] строго положительна при всех x [$8712$] R, но f(x) > 0 только при x > 0. Дополнительно должно выполняться условие
б) Неверно. Например, для функции f(x) = (x-3)[sup]3[/sup] её производная f '(x) = 3(x-3)[sup]2[/sup] равна нулю в точке x = 3, но функция f(x) не имеет в этой точке максимума или минимума (только перегиб). Дополнительно должно выполняться условие f "(3) [$8800$] 0.
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