Question

Tìm giá trị nhỏ nhất của |x-1|+|x-2|+|x-3|+... +|x-2020|

Answer 1

`|x-1|+|x-2|+|x-3|+....+|x-2020|`

`=(|x-1|+|x-2020|)+(|x-2|+|x-2019|)+....+(|x-1000|+|x-1001|)`

Áp dụng bđt `|A|+|B|>=|A+B|`

`=>|x-1|+|x-2020|=|x-1|+|2020-x|>=|x-1+2020-x|=2019`

Tương tự:

`|x-2|+|x-2019|>=2017`

`.................................`

`|x-1000|+|x-1001|>=1`

`=>|x-1|+|x-2|+|x-3|+....+|x-2020|>=2019+2017+....+1`

`=>|x-1|+|x-2|+|x-3|+....+|x-2020|>=((2019+1).2019)/2=2039190`

Dấu "=" xảy ra khi `{((x-1)(2020-x)>=0),((x-2)(2019-x)>=0),(.........),((x-1000)(1001-x)>=0):}`

`<=>{((x-1)(x-2020)<=0),((x-2)(x-2019)<=0),(.........),((x-1000)(x-1001)<=0):}`

`<=>{(1<=x<=2020),(2<=x<=2019),(.........),(1000<=x<=1001):}`

`<=>1000<=x<=1001`