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Консультация № 62898
16.11.2006, 16:56
0.00 руб.
0
2
1
Объясните как найти точки экстремума для функции
z=6xy-9(x^2)-9(y^2)+4x+4y
z=6xy-9(x^2)-9(y^2)+4x+4y
Обсуждение
16.11.2006, 18:46
общий
это ответ
Здравствуйте, Solnse!
Решение.
Находим частные производные первого порядка:
Dz/Dx=6y-18x+4, Dz/Dy=6x-18y+4.
Воспользовавшись необходимыми условиями экстремума, находим стационарные точки, т. е. решаем систему двух уравнений
6y-18x+4=0 и 6x-18y+4=0, откуда x=1/3, y=1/3; M (1/3; 1/3).
Находим значения частных производных второго порядка в точке М:
A=D^2 z/Dx^2=-18, B=D^2 z/Dy^2=-18, C=D^2 z/(DxDy)=6
и составляем дискриминант Д=AC-B^2=(-18)*(-18)-6^2=324-36=288. Поскольку Д>0, A<0, то данная функция в точке M (1/3; 1/3) имеет максимум. Значение функции в этой точке z(1/3; 1/3)=6*(1/3)*(1/3)-9*(1/3)^2-9*(1/3)^2+4*(1/3)+4*(1/3)=1 1/3 (проверьте).
Ответ: 1 1/3.
С уважением,
Mr. Andy.
Решение.
Находим частные производные первого порядка:
Dz/Dx=6y-18x+4, Dz/Dy=6x-18y+4.
Воспользовавшись необходимыми условиями экстремума, находим стационарные точки, т. е. решаем систему двух уравнений
6y-18x+4=0 и 6x-18y+4=0, откуда x=1/3, y=1/3; M (1/3; 1/3).
Находим значения частных производных второго порядка в точке М:
A=D^2 z/Dx^2=-18, B=D^2 z/Dy^2=-18, C=D^2 z/(DxDy)=6
и составляем дискриминант Д=AC-B^2=(-18)*(-18)-6^2=324-36=288. Поскольку Д>0, A<0, то данная функция в точке M (1/3; 1/3) имеет максимум. Значение функции в этой точке z(1/3; 1/3)=6*(1/3)*(1/3)-9*(1/3)^2-9*(1/3)^2+4*(1/3)+4*(1/3)=1 1/3 (проверьте).
Ответ: 1 1/3.
С уважением,
Mr. Andy.
Об авторе:
Facta loquuntur.
Facta loquuntur.
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