x=a²-2b+1,y=b²-2c+2,z=c²-2b+3,求x+y+z的取值范围

x=a²-2b+1,y=b²-2c+2,z=c²-2b+3,求x+y+z的取值范围
x=a²-2b+1,y=b²-2c+2,z=c²-2b+3,求x+y+z的取值范围

x=a²-2b+1,y=b²-2c+2,z=c²-2b+3,求x+y+z的取值范围
x + y + z = a² - 2b + 1 + b² -2c + 2 + c² - 2b + 3
= a² + b² - 4b + 4 - 4 + c² - 2c + 1 - 1 + 1 + 2 + 3
= a² + (b-2)² + (c-1)² + 1

显然, x + y + z >= 1, 当 a = 0, b = 2, c = 1 时, 取等号

取值范围
大于或等于1