Showing sup{f(x)} - inf{f(x)} >= sup{|f(x)|} - inf{|f(x)|}

[jdinatale]

Hello, below I have typed the problem and drew a diagram to help me think about the problem.

It seems intuitive to me that $M \leq M'$ since M' might be the absolute value of the most extreme negative f value that is greater in magnitude than the most extreme positive f value. Also, $m \leq m'$ since $m' \geq 0$ and m could be negative.

Is this type of problem best handled with cases such as Case 1: $f(x) \leq 0$?

 

[Bacle2]

I think working out the individual cases may do it. It seems to come down to showing
that:

A-B ≥ |A|-|B|

So that you get equality. Only real case is when A,B have different signs. Maybe
you can show it geometrically, using |X| asthe distance from X to a fixed value
(thinking of 0 in the real line).