Which would be larger x² or x³/3? Is this always the case?

Answer 1

Great question. Unfortunately your question does not have a single answer. Sometimes x² is the greater and sometimes it is smaller. Let's investigate when that might be. Let's solve for all of the times when x² is greater than x³/3.

Starting with: X² > X³/3

we see that we can divide both sides by X². We don't have to worry about signs when we do that because X² is always positive.

1 > X/3

3 > X.

So for all values less that 3, X² is greater than X³/3. You can verify that the answer makes sense by graphing each side as a function of x.