Question

What are the absolute extrema of f(x)=|x-1|-3 on (0,4]?

Answer 1

One abs. Minima only: (1,-3)
One abs. maximum at (4,0)

Explanation:

You can just simply graph it. However, I assume you are asking for a mathematic approach...
First, split it into a piecewise function
y1=x-4 when 4>=x>=1
y2=-x-2 when 0<x<1
First derivative for y1 is always 1 (positive), meaning it is always increasing. Thus, the left most endpoint is the minima (x=1). The right endpoint is maxima at x=4.
First derivative for y2 is always -1 (negative), meaning it's always decreasing. Here you can find a minima at x=0 and maxima x=1. But when comparing these points to the ones we got from y1, both of them from y2 are not the largest or the smallest. Thus render them useless.
Plug x=1 back in and find y=-3.
Plug x=4 back in and find y=0.