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Консультация № 111999
02.12.2007, 14:47
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Исследовать данные функции методами дифферинциального исчисления и построить их графики. При исследовании функции нужно найти интервалы возрастания и убывания и точки эктремума функции, интервалы выпуклости и вогнутости и точки перегиба графика функции. у=1/3Х3-2Х2+3Х-1
Спасибо!
Спасибо!
Обсуждение
Неизвестный
02.12.2007, 14:56
общий
это ответ
Здравствуйте, Николаева Ольга Николаевна!
Берём производную y‘ = x<SUP>2</SUP>-4x+3. y‘<0 ⇔ x ∈ (1,3) соответствует убыванию функции; y‘=0, т.е. x=1 и x=3 - экстремумам; x<1 и x>3 - возрастанию функции.
Берём вторую производную, y‘‘ = 2x-4. y‘‘=0 ⇔ x=2 - точка перегиба.
y‘‘>0 ⇔ x ∈ (2,+∞) соответствует выпуклости вниз / вогнутости, y‘‘ < 0 ⇔ x < 2 - выпуклости вверх.
Вспомним про экстремумы, y‘‘(1)=-2< ⇒ x=1 - максимум, y‘‘(3)>0 ⇒ x=3 - минимум.
После такого исследования нетрудно построить график: в точке (1,1/3) - максимум, в точке (2,-1/3) - перегиб, в точке (3,-1) - минимум, затем кривая резко уходит вверх.
Удачи!<p><fieldset style=‘background-color:#EFEFEF; width:80%; border:#777777 1px solid; padding:10px;‘ class=fieldset><font color=#777777><i>Дополнено из мини-форума.</i>
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</font><font color=#777777 size=1><b>• Отредактировал: <a href=/info/user/118729 target=_blank>Агапов Марсель</a></b> (Специалист)
<b>• Дата редактирования:</b> 06.12.2007, 16:49</font></fieldset>
Берём производную y‘ = x<SUP>2</SUP>-4x+3. y‘<0 ⇔ x ∈ (1,3) соответствует убыванию функции; y‘=0, т.е. x=1 и x=3 - экстремумам; x<1 и x>3 - возрастанию функции.
Берём вторую производную, y‘‘ = 2x-4. y‘‘=0 ⇔ x=2 - точка перегиба.
y‘‘>0 ⇔ x ∈ (2,+∞) соответствует выпуклости вниз / вогнутости, y‘‘ < 0 ⇔ x < 2 - выпуклости вверх.
Вспомним про экстремумы, y‘‘(1)=-2< ⇒ x=1 - максимум, y‘‘(3)>0 ⇒ x=3 - минимум.
После такого исследования нетрудно построить график: в точке (1,1/3) - максимум, в точке (2,-1/3) - перегиб, в точке (3,-1) - минимум, затем кривая резко уходит вверх.
Удачи!<p><fieldset style=‘background-color:#EFEFEF; width:80%; border:#777777 1px solid; padding:10px;‘ class=fieldset><font color=#777777><i>Дополнено из мини-форума.</i>
-----
</font><font color=#777777 size=1><b>• Отредактировал: <a href=/info/user/118729 target=_blank>Агапов Марсель</a></b> (Специалист)
<b>• Дата редактирования:</b> 06.12.2007, 16:49</font></fieldset>
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