Question

the given expression is |x-1|+|x-2|+|x-3|+....+|x-2019|+|x-2020|. determine the greatest interval of real numbers [a,b] on which the given expression has a constant value k for all x € [a,b] what is the value pf k? please give me full explanation ​

Answer 1

Answer:

Use that |a−b|=|b−a||a−b|=|b−a| and notice that we have by triangle inequality

|x−y|+|y−z|≥|x−z||x−y|+|y−z|≥|x−z|

So

|x−1|+|x−2020||x−2|+|x−2019||x−1010|+|x−1011|=|1−x|+|x−2020|≥2019=|2−x|+|x−2019|≥2017⋮=|1010−x|+|x−1011|≥1|x−1|+|x−2020|=|1−x|+|x−2020|≥2019|x−2|+|x−2019|=|2−x|+|x−2019|≥2017⋮|x−1010|+|x−1011|=|1010−x|+|x−1011|≥1

Thus

f(x)≥505⋅2020=10102f(x)≥505⋅2020=10102

To achive that value we have to have equality case everywhere. Say a<ba<b, since