若x,y,z均为非负数,且满足x−1=y+1/2=z−23,则x2+y2+z2可取得的最小值为_.

问题描述:

若x,y,z均为非负数,且满足x−1=

y+1
2
z−2
3
,则x2+y2+z2可取得的最小值为______.

令x-1=y+12=z−23=t,则x=t+1,y=2t-1,z=3t+2,于是x2+y2+z2=(t+1)2+(2t-1)2+(3t+2)2=t2+2t+1+4t2+1-4t+9t2+4+12t=14t2+10t+6,∵x,y,z均为非负数,∴x-1≥-1,y+12≥12,z−23≥-23,∵x-1=y+12=t,∴y≥1...