Question

What is the maximum and minimum of `abs(x)+abs(x-1)+abs(x-2)` ?

Answer 1

This expression has no maximum value. It attains a minimum value `2`.

Explanation:

Note that each of the functions, `abs(x)`, `abs(x-1)` and `abs(x-2)` is V-shaped, with a minimum where the expression inside the `abs(...)` is zero.

Hence, the minimum value of the sum will occur when `x=1`:

`abs(color(blue)(1)) + abs(color(blue)(1)-1) + abs(color(blue)(1)-2) = abs(1)+abs(0)+abs(-1) = 1+0+1 = 2`

When `x >=2` we have:

`abs(x) + abs(x-1) + abs(x-2) = x+(x-1)+(x-2) = 3x-3`

which increases without upper bound as `x` increases.

graph{abs(x)+abs(x-1)+abs(x-2) [-9.24, 10.76, -2.74, 7.26]}