Question

X－1的绝对值+X－2的绝对值+X－3的绝对值+X－4的绝对值+X－5的绝对值+X－6的绝对值的最小值为什么是9

Answer 1

当x<=1时，f(x)=-(x-1)-(x-2)-(x-3)-(x-4)-(x-5)-(x-6)=-6x+21
当x=1，取最小f(x)=15,
当1<x<=2时，f(x)=(x-1)-(x-2)-(x-3)-(x-4)-(x-5)-(x-6)=-4x+20
当x=2，取最小f(x)=12,
当2<x<=3时，f(x)=(x-1)+(x-2)-(x-3)-(x-4)-(x-5)-(x-6)=-2x+15
当x=3时，取最小f(x)=9,
当3<x<=4时，f(x)=(x-1)+(x-2)+(x-3)-(x-4)-(x-5)-(x-6)=9
当x=4时，取最小f(x)=9,
当4<x<=5时，f(x)=(x-1)+(x-2)+(x-3)+(x-4)-(x-5)-(x-6)=2x+1
当x=5时，取最小f(x)=11,
当5<x<=6时，f(x)=(x-1)+(x-2)+(x-3)+(x-4)+(x-5)-(x-6)=4x-9
当x=6时，取最小f(x)=15,
当x>=6时，f(x)=(x-1)+(x-2)+(x-3)+(x-4)+(x-5)+(x-6)=6x-21
综上 当3<=x<=4时 取最小值9

Answer 2

建议你用几何画板软件，建立函数f(x)min=|x-1|+|x-2|+|x-3|+|x-4|+|x-5|+|x-6|，模拟取值即可观察。