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Консультация № 185580
11.03.2012, 16:51
63.12 руб.
0
1
1
Здравствуйте! У меня возникли сложности с таким вопросом:
Прошу подробно расписать решение по образцу https://rfpro.ru/upload/7685
Задание
Найти проекцию точки (0,0)[$8712$]R2 на полуплоскость аx + by [$8805$] c, если a= 1, b= -3, c=1.
Прошу подробно расписать решение по образцу https://rfpro.ru/upload/7685
Задание
Найти проекцию точки (0,0)[$8712$]R2 на полуплоскость аx + by [$8805$] c, если a= 1, b= -3, c=1.
Обсуждение
11.03.2012, 21:07
общий
это ответ
Здравствуйте, Посетитель - 356695!
Найти проекцию точки (0,0) ∈R2 на полуплоскость аx+ by >= c, если a= 1, b= -3, c=1.
f=x2+y2→min
x-3y≥1
Введем дополнительные переменные
x-3y-z=1, z≥0
L=λ0(x2+y2)+λ1(x-3y-z-1)
Lx=2λ0x+λ1=0
Ly=2λ0y-3λ1=0
Lz=-λ1z=0
Lλ1= x-3y-z-1=0
1 случай
λ0=0 λ1≠0
2λ1=0 λ1=0
2 случай
λ0≠0 λ0=1
2x+λ1=0 x=- λ1/2
3y-3λ1=0 y=3/2λ1
-λ1z=0
-λ1=0 → x=0, y=0
x-3y-z-1=0
z=-1
z=0
x-3y=1
- λ1/2-9/2λ1=1
-5λ1=1
λ1=-1/5
x=1/10
y=-3/10
Дост. условие
Lxx=2
Lyy=2
Lzz=0
d2L=2(dx2+dy2)>0
В точке (1/10; 3/10) минимальная проекция
Найти проекцию точки (0,0) ∈R2 на полуплоскость аx+ by >= c, если a= 1, b= -3, c=1.
f=x2+y2→min
x-3y≥1
Введем дополнительные переменные
x-3y-z=1, z≥0
L=λ0(x2+y2)+λ1(x-3y-z-1)
Lx=2λ0x+λ1=0
Ly=2λ0y-3λ1=0
Lz=-λ1z=0
Lλ1= x-3y-z-1=0
1 случай
λ0=0 λ1≠0
2λ1=0 λ1=0
2 случай
λ0≠0 λ0=1
2x+λ1=0 x=- λ1/2
3y-3λ1=0 y=3/2λ1
-λ1z=0
-λ1=0 → x=0, y=0
x-3y-z-1=0
z=-1
z=0
x-3y=1
- λ1/2-9/2λ1=1
-5λ1=1
λ1=-1/5
x=1/10
y=-3/10
Дост. условие
Lxx=2
Lyy=2
Lzz=0
d2L=2(dx2+dy2)>0
В точке (1/10; 3/10) минимальная проекция
5
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