Question

How to find the max and minimum of `f(x)= abs(x-1 )+ 2abs(x+5) + 3abs(x-4)` using derivatives?

Answer 1

See below.

Explanation:

We can do that using the so called sub-derivative or sub-gradient. See https://en.wikipedia.org/wiki/Subderivative
This tool is used in non-smooth optimization. It is a derivative generalization for locally convex functions.