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Консультация № 185775
08.04.2012, 11:52
62.88 руб.
0
1
1
Уважаемые эксперты! Пожалуйста, ответьте на вопрос:
Найти наибольшее и наименьшее значения функции z=f(x,y) в замкнутой области D, заданной указанными линиями:
z=x^3+y^3-3xy, x=0, x=2, y=0, y=3
Найти наибольшее и наименьшее значения функции z=f(x,y) в замкнутой области D, заданной указанными линиями:
z=x^3+y^3-3xy, x=0, x=2, y=0, y=3
Обсуждение
08.04.2012, 13:24
общий
это ответ
Здравствуйте, Aleksandrkib!
Приравниваем частные производные к 0:
z'x=3x^2-3y=0
z'y=3y^2-3x=0
Из решения системы находим x=y=1, z(1,1)=1+1-3=-1
Рассмотрим границы области:
1) х=0 (0<=y<=3): z=y^3, zmin=0, zmax=27
2) х=2 (0<=y<=3): z=y^3-6y+8, zmin=8-4*sqrt{2}~2,34 (y=sqrt{2}), zmax=17 (y=3)
3) y=0 (0<=x<=2): z=x^3, zmin=0, zmax=8
4) y=3 (0<=x<=2): z=x^3-9x+27, zmin=27-6*sqrt{3}~16,61 (x=sqrt{3}), zmax=27 (x=0)
Ответ: zmin=-1, zmax=27
Приравниваем частные производные к 0:
z'x=3x^2-3y=0
z'y=3y^2-3x=0
Из решения системы находим x=y=1, z(1,1)=1+1-3=-1
Рассмотрим границы области:
1) х=0 (0<=y<=3): z=y^3, zmin=0, zmax=27
2) х=2 (0<=y<=3): z=y^3-6y+8, zmin=8-4*sqrt{2}~2,34 (y=sqrt{2}), zmax=17 (y=3)
3) y=0 (0<=x<=2): z=x^3, zmin=0, zmax=8
4) y=3 (0<=x<=2): z=x^3-9x+27, zmin=27-6*sqrt{3}~16,61 (x=sqrt{3}), zmax=27 (x=0)
Ответ: zmin=-1, zmax=27
Форма ответа
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