Question

(y-1)^2+(x+y-3)^2+(2x+y-6)^2的最小值是多少

Answer 1

(y-1)^2+(x+y-3)^2+(2x+y-6)^2=5x^2+6xy+3y^2-30x-20y+46
=3y^2+(6x-20)y+(5x^2-30x+46)
=3(y+x-10/3)^2+(5x^2-30x+46)-3(x-10/3)^2
=3(x+y-10/3)^2+2x^2-(30-20)x+46-100/3
=3(x+y-10/3)^2+2(x-5/2)^2+38/3-25/2
=3(x+y-10/3)^2+2(x-5/2)^2+1/6>=1/6
上式中等号在x=5/2,y=10/3-5/2=5/6时成立。
故，当x=5/2,y=5/6时，(y-1)^2+(x+y-3)^2+(2x+y-6)^2取得最小值1/6.

Answer 2

大概是三个都等于零最小,但第二和第三不能同时等于零,因此先令Y=1
(X-2)^2+(2X-5)^2=5X^2-24X+29=5(X-12/5)^2+29-144/5>=1/5.(此时X=12/5)

Answer 3

解:
y=1
s=(y-1)^2+(x+y-3)^2+(2x+y-6)^2
=(x-2)^2+(2x-5)^2
=5(x-2.4)^2+0.2
x=2.4,s=0.2

x=3
(y-1)^2+(x+y-3)^2+(2x+y-6)^2
=(y-1)^2+y^2+y^2
=3(y-1/3)^2+2/3≥2/3>0.2

y=1.01,x=2.394,s=0.20412>0.2
y=0.99,s=0.19612<0.2
设y=1-a,a>0
s=(y-1)^2+(x+y-3)^2+(2x+y-6)^2
=(1-a-1)^2+(x+1-a-3)^2+(2x+1-a-6)^2
=5[x-0.6(a+4)]^2+[6*(a-1/6)^2+5/6]/5
x=0.6(a+4)
s最小值k=[6*(a-1/6)^2+5/6]/5
a=1/6,k最小值=(5/6)/5=1/6
x=0.6(a+4)=2.5,y=1-1/6=5/6,s=1/6
检验:
(y-1)^2+(x+y-3)^2+(2x+y-6)^2
=(5/6-1)^2+(2.5+5/6-3)^2+(2*2.5+5/6-6)^2
=1/36+1/9+1/36
=1/6
答:x=2.5,y=5/6,(y-1)^2+(x+y-3)^2+(2x+y-6)^2 最小值=1/6