This is a funny little puzzle!

If I look at your function it is built from six components, each component being the absolute amount of a linear function. Now let us look at a component function, for example

y1 = |-22-16x|

This function has two linear legs, a descending leg on the left side and and a rising leg on the right side. For x = -1.375 it takes the value zero.

The same is true for the other five component functions. Each of them is built from two legs and has a low point, where it takes the value 0. I conclude that the function y that you have exposed, has six

points where it is not differentiable. Newton-Raphson cannot be used.

However, your function has another property, which makes a solution easy. With the exception of the low points of the six component functions your function can be built by adding six linear functions. Therefore it is linear itself between two low points. As a consequence one of the low points of the component function is the minimum for which you are looking.

Therefore you just need to calculate x1, x2, x3, x4, x5, x6 representing the low points of the component functions and then find out for which of those the function

y = |-22-16x|+|44-22x|+|21-25x|+|40-22x|+|6-11x|+|12-4x|

takes the lowest value.