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Консультация № 159466
03.02.2009, 09:22
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Добрый день, уважаемые эксперты!
Помогите пожалуйста разобраться: требуется исследовать функцию y=(x^3)/(3*(3-x^2)) методами дифференциального исчисления. (Найти: область существования, точки разрыва, точки экстремума, интервалы возрастания и убывания, точки перегиба, интервалы выпуклости и вогнутости, и т.д.). Меня смущает знаменатель 3-x^2.
Заранее благодарю.
Помогите пожалуйста разобраться: требуется исследовать функцию y=(x^3)/(3*(3-x^2)) методами дифференциального исчисления. (Найти: область существования, точки разрыва, точки экстремума, интервалы возрастания и убывания, точки перегиба, интервалы выпуклости и вогнутости, и т.д.). Меня смущает знаменатель 3-x^2.
Заранее благодарю.
Обсуждение
Неизвестный
05.02.2009, 06:33
общий
это ответ
Здравствуйте, Vasja21!
1) Обл. определения: x не равен +/-sqrt(3) =>
D(f):(-бескон;-sqrt(3))в объединении с (-sqrt(3);sqrt(3))в объединении с (sqrt(3);+беск.)
2) х=-sqrt(3), x=sqrt(3) - точки разрыва.
lim_x->(-sqrt(3)+0)[(x^3)/(3*(3-x^2))]=-бесконеч,
lim_x->(sqrt(3)+0)[(x^3)/(3*(3-x^2))]=-бесконеч,
lim_x->(-sqrt(3)-0)[(x^3)/(3*(3-x^2))]=бесконеч,
lim_x->(sqrt(3)-0)[(x^3)/(3*(3-x^2))]=бесконеч
=> х=-sqrt(3), x=sqrt(3) - точки разрыва второго рода.
3) y'=[(3x^2)*3*(3-x^2)-x^3*(-6x)]/9(3-x^2)^2=(9x^2-x^4)/3(3-x^2)^2
y'=0 => (9x^2-x^4)=0 => x^2(9-x^2)=0 => x1=0, x2=-3, x3=3
Найдем промежутки возрастания, убывания: (-беск;-3), (3;+беск) - функция у убывает,
на промежутках (-3;0) и (0;3) -функция у возрастает.
х=-3 точка локального минимума, х=3 - точка локального максимума
4) y''=(18x-4x^3)3(3-x^2)^2-(9x^2-x^4)*(12x(3-x^2))/9(3-x^2)^4=
=3(18x-4x^3)(3-x^2)-x^2(9-x^2)*(12x)/9(3-x^2)^3=
=162x-54x^3-36x^3+12x^5-9x^2+x^4/9(3-x^2)^3
y''=0 => 162x-18x^3+12x^5-9x^2+x^4=0 Действительный корень 1 - х=0
(-беск; 0)- функция вогнута, (0;+беск)- функц. выпукла
=> x=0 - точка перегиба.
1) Обл. определения: x не равен +/-sqrt(3) =>
D(f):(-бескон;-sqrt(3))в объединении с (-sqrt(3);sqrt(3))в объединении с (sqrt(3);+беск.)
2) х=-sqrt(3), x=sqrt(3) - точки разрыва.
lim_x->(-sqrt(3)+0)[(x^3)/(3*(3-x^2))]=-бесконеч,
lim_x->(sqrt(3)+0)[(x^3)/(3*(3-x^2))]=-бесконеч,
lim_x->(-sqrt(3)-0)[(x^3)/(3*(3-x^2))]=бесконеч,
lim_x->(sqrt(3)-0)[(x^3)/(3*(3-x^2))]=бесконеч
=> х=-sqrt(3), x=sqrt(3) - точки разрыва второго рода.
3) y'=[(3x^2)*3*(3-x^2)-x^3*(-6x)]/9(3-x^2)^2=(9x^2-x^4)/3(3-x^2)^2
y'=0 => (9x^2-x^4)=0 => x^2(9-x^2)=0 => x1=0, x2=-3, x3=3
Найдем промежутки возрастания, убывания: (-беск;-3), (3;+беск) - функция у убывает,
на промежутках (-3;0) и (0;3) -функция у возрастает.
х=-3 точка локального минимума, х=3 - точка локального максимума
4) y''=(18x-4x^3)3(3-x^2)^2-(9x^2-x^4)*(12x(3-x^2))/9(3-x^2)^4=
=3(18x-4x^3)(3-x^2)-x^2(9-x^2)*(12x)/9(3-x^2)^3=
=162x-54x^3-36x^3+12x^5-9x^2+x^4/9(3-x^2)^3
y''=0 => 162x-18x^3+12x^5-9x^2+x^4=0 Действительный корень 1 - х=0
(-беск; 0)- функция вогнута, (0;+беск)- функц. выпукла
=> x=0 - точка перегиба.
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