Hi. I had the problem below for homework. I think it is true, but I don't know why. Can anyone give me a reason why this is true, because I need to provide an explination. Thanks!
True or False:
If a<b then a3 <b3
I assume you mean: if then .
It is actually true, but needs a fair amount of work to demonstrate.
First, you can show it's true when both and are positive, as follows.
You know that if then if then .
Okay, so implies (putting in the above).
Then implies (putting in the above).
So you've shown implies .
Do the same (similar) thing to prove for .
You go through a similar process when and are both negative, and again when is negative and positive.
An alternative approach makes use of the fact that the cube function is monotonically increasing - but that's really on a calculus course.