求实数x、y的值,使得(y-1)2+(x+y-3)2+(2x+y-6)2达到最小值.

问题描述:

求实数x、y的值,使得(y-1)2+(x+y-3)2+(2x+y-6)2达到最小值.

a=y-1,b=x+y-3,c=2x+y-6,可得a-2b+c=-1,则原式=a2+b2+c2=a2+14b2+14b2+14b2+14b2+c2≥16(a-12b-12b-12b-12b+c)2=16(a-2b+c)2=16,取等条件a=-12b=c,即y-1=-12(x+y-3)=2x+y-6,y-1=-12(x+y-3)y-1=2x+y-6,...