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Консультация № 178469
19.05.2010, 02:16
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добрый вечер!помогите пожалуйста с решением - нужно провести полное исследование функции и построить ее график
y=(x^3-32)/x^2
y=(x^3-32)/x^2
Обсуждение
Неизвестный
19.05.2010, 04:32
общий
это ответ
Здравствуйте, инна сергеевна ахматова.
y=(x^3-32)/x^2 =x-32/x2
y(-x) [$8800$] y(x) [$8800$] -y(x)
Функция не является четной или нечетной.
Область определения: (-[$8734$];0)[$8746$](0;[$8734$]+)
y'=1-32*(-2)/x3=(x3+64)/x3
(x3+64)/x3=0 -> x3+64= (x+4)*(x2-4*x+16)=0 -> x= -4
y(-4)= -6
функция возрастает на (-[$8734$];-4), (0;[$8734$]+)
функция убывает на (-4;0)
x= -4 - точка локального максимума
y''=(1+64/x3)'= -192/x4 < 0 при любых x
интервалы выпуклости: (-[$8734$];0),(0;[$8734$]+)
точек перегиба нет
Нули функции
y=(x3-32)/x2=0 -> x3-32=0
корень один x=321/3=3.1748...
Асимптоты графика функции
limx->0y(x)= -[$8734$] -> x=0 - вертикальная асимптота
наклонная асимптота:
y=k*x+b
k=limx->[$8734$] y(x)/x=limx->[$8734$](1-32/x3)=1
b=limx->[$8734$] (y-k*x)=limx->[$8734$](-32/x2)=0
y=x - наклонная асимптота
y=(x^3-32)/x^2 =x-32/x2
y(-x) [$8800$] y(x) [$8800$] -y(x)
Функция не является четной или нечетной.
Область определения: (-[$8734$];0)[$8746$](0;[$8734$]+)
y'=1-32*(-2)/x3=(x3+64)/x3
(x3+64)/x3=0 -> x3+64= (x+4)*(x2-4*x+16)=0 -> x= -4
y(-4)= -6
функция возрастает на (-[$8734$];-4), (0;[$8734$]+)
функция убывает на (-4;0)
x= -4 - точка локального максимума
y''=(1+64/x3)'= -192/x4 < 0 при любых x
интервалы выпуклости: (-[$8734$];0),(0;[$8734$]+)
точек перегиба нет
Нули функции
y=(x3-32)/x2=0 -> x3-32=0
корень один x=321/3=3.1748...
Асимптоты графика функции
limx->0y(x)= -[$8734$] -> x=0 - вертикальная асимптота
наклонная асимптота:
y=k*x+b
k=limx->[$8734$] y(x)/x=limx->[$8734$](1-32/x3)=1
b=limx->[$8734$] (y-k*x)=limx->[$8734$](-32/x2)=0
y=x - наклонная асимптота
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