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Консультация № 185283
23.01.2012, 23:24
62.27 руб.
25.01.2012, 10:23
0
2
2
Здравствуйте, уважаемые эксперты! Прошу вас ответить на следующий вопрос: Найдите локальные экстремумы функции
Обсуждение
24.01.2012, 01:25
общий
это ответ
Здравствуйте, Посетитель - 383833!
Необходимым условием локального экстремума является равенство 0 частных производных.
dz/dx=3x^2+y^2+3y
dz/dy=2xy+3x
2xy+3x=0[$8658$]x(2y+3)=0[$8658$]x=0 или y=-1.5
Если x=0, dz/dx=0 если y=0 или y=-3
Если y=-1.5, dz/dx=0 если 3x^2+2,25-4,5=0
3x^2=2,25
x=[$8730$]0.75
Стационарными точками являются (0,0), (0,-3) ([$8730$]0.75, -1.5) (-[$8730$]0.75, -1.5)
Для определения достаточных условий надо рассмотреть знак выражения
Для ст. точки (0,0) D <0 и экстремума нет
Для ст. точки (0,-3)D <0 и экстремума нет
Для ст. точки ([$8730$]0.75, -1.5))D=
и имеем минимум.
Для ст. точки (-[$8730$]0.75, -1.5)) D=
и имеем максимум.
Необходимым условием локального экстремума является равенство 0 частных производных.
dz/dx=3x^2+y^2+3y
dz/dy=2xy+3x
2xy+3x=0[$8658$]x(2y+3)=0[$8658$]x=0 или y=-1.5
Если x=0, dz/dx=0 если y=0 или y=-3
Если y=-1.5, dz/dx=0 если 3x^2+2,25-4,5=0
3x^2=2,25
x=[$8730$]0.75
Стационарными точками являются (0,0), (0,-3) ([$8730$]0.75, -1.5) (-[$8730$]0.75, -1.5)
Для определения достаточных условий надо рассмотреть знак выражения
Для ст. точки (0,0) D <0 и экстремума нет
Для ст. точки (0,-3)D <0 и экстремума нет
Для ст. точки ([$8730$]0.75, -1.5))D=
и имеем минимум.Для ст. точки (-[$8730$]0.75, -1.5)) D=
и имеем максимум.
Неизвестный
24.01.2012, 01:49
общий
это ответ
Здравствуйте, Посетитель - 383833!
1. Найдём стационарные точки. Для этого вычислим первые частные производные, приравняем их к нулю и решим получившуюся систему уравнений.
Получаем четыре стационарные точки:
2. Проверим, в каких стационарных точках выполняется достаточное условие локального экстремума.
В точках
и 
нет экстремума:
Экстремумы в точках
и 
:
Так как
, то 
точка максимума.
Так как
, то 
точка минимума.
Ответ:
- точка максимума; 
- точка минимума.
1. Найдём стационарные точки. Для этого вычислим первые частные производные, приравняем их к нулю и решим получившуюся систему уравнений.
Получаем четыре стационарные точки:
2. Проверим, в каких стационарных точках выполняется достаточное условие локального экстремума.
В точках
и нет экстремума:Экстремумы в точках
и :Так как
, то точка максимума.Так как
, то точка минимума.Ответ:
- точка максимума; - точка минимума.
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