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Консультация № 67588
18.12.2006, 21:25
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Дорогие эксперты! помоги мне пожалуста!
2.Найти наибольшее и наименьшее значение функции.Найти наименьшее и наибольшее значение функции в области,отражанной задаными линиями:
функция Z=(X-Y)*(X+Y);
область (X^2)-(Y^2)(меньше или равно) 3
С уважением, Олимп.
2.Найти наибольшее и наименьшее значение функции.Найти наименьшее и наибольшее значение функции в области,отражанной задаными линиями:
функция Z=(X-Y)*(X+Y);
область (X^2)-(Y^2)(меньше или равно) 3
С уважением, Олимп.
Обсуждение
21.12.2006, 11:26
общий
это ответ
Здравствуйте, Olimp!
Выполним преобразование z=(x-y)*(x+y)=x^2-y^2.
Находим стационарные точки данной функции. Имеем
∂z/∂x=2*x, ∂z/∂y=-2*y.
В силу необходимых условий экстремума находим
x=0, y=0, z (0; 0)=0.
Исследуем заданную функцию на условный экстремум, если x и y связаны соотношением x^2-y^2=3. Рассмотрим функцию Лагранжа u=x^2-y^2+λ*(x^2-y^2-3). Ее частные производные
∂u/∂x=2*x+2*λ*x, ∂u/∂y=-2*y-2*λ*y.
Для определения x, y и λ получаем систему уравнений
2*x+2*λ*x=0,
-2*y-2*λ*y=0,
x^2-y^2=3,
откуда находим
x1=-√3, y1=0, λ1=-1, z1=3,
x2=√3, y2=0, λ2=-1, z2=3.
Итак,
zmax=z (-√3; 0)= z (-√3; 0)=3, zmin=z (0; 0)=0.
С уважением,
Mr. Andy.
Выполним преобразование z=(x-y)*(x+y)=x^2-y^2.
Находим стационарные точки данной функции. Имеем
∂z/∂x=2*x, ∂z/∂y=-2*y.
В силу необходимых условий экстремума находим
x=0, y=0, z (0; 0)=0.
Исследуем заданную функцию на условный экстремум, если x и y связаны соотношением x^2-y^2=3. Рассмотрим функцию Лагранжа u=x^2-y^2+λ*(x^2-y^2-3). Ее частные производные
∂u/∂x=2*x+2*λ*x, ∂u/∂y=-2*y-2*λ*y.
Для определения x, y и λ получаем систему уравнений
2*x+2*λ*x=0,
-2*y-2*λ*y=0,
x^2-y^2=3,
откуда находим
x1=-√3, y1=0, λ1=-1, z1=3,
x2=√3, y2=0, λ2=-1, z2=3.
Итак,
zmax=z (-√3; 0)= z (-√3; 0)=3, zmin=z (0; 0)=0.
С уважением,
Mr. Andy.
Об авторе:
Facta loquuntur.
Facta loquuntur.
Форма ответа
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