Prove that $f(x)=x^3$ increases everywhere.

Recall that a function increases on an interval $I$ when for every $x_1$ and $x_2$ in $I$, if <Sign in to see all the formulas>, then <Sign in to see all the formulas>.

For this particular problem we are wanting to show that <Sign in to see all the formulas> increases everywhere. When we say everywhere we really mean the interval that contains every number. So now let's assume we pick any two numbers $x_1$ and $x_2$ such that <Sign in to see all the formulas>. We have to deal with three cases:

• $x_1$ and $x_2$ are both positive,
• $x_1$ and $x_2$ are both negative,
• and <Sign in to see all the formulas> and <Sign in to see all the formulas>.

(Note that we don't have the case <Sign in to see all the formulas> and <Sign in to see all the formulas> because we make sure we picked $x_1$ and $x_2$ such that <Sign in to see all the formulas>.) In the first two cases note that <Sign in to see all the formulas> and in the third case <Sign in to see all the formulas>.

We are going to prove the first two cases together knowing that for both of them <Sign in to see all the formulas>. First, we know that

<Sign in to see all the formulas>

Second, we know that

<Sign in to see all the formulas>

Third, we know that because <Sign in to see all the formulas> that

<Sign in to see all the formulas>

Now by putting our inequalities together we see that

<Sign in to see all the formulas>

and therefore <Sign in to see all the formulas> when <Sign in to see all the formulas>.

Now for our third case we have <Sign in to see all the formulas>. In this case <Sign in to see all the formulas> and <Sign in to see all the formulas> so that <Sign in to see all the formulas>.

Finally, what happens when <Sign in to see all the formulas>? In that case either $x_1$ is zero or $x_2$ is zero. If $x_1=0$ then $x_2$ is positive by the assumption that <Sign in to see all the formulas>. Therefore <Sign in to see all the formulas>. When $x_2=0$ then $x_1$ is negative by the assumption that <Sign in to see all the formulas>. Therefore, <Sign in to see all the formulas>.

We have now proven that for all <Sign in to see all the formulas> that <Sign in to see all the formulas>. Therefore, <Sign in to see all the formulas> increases everywhere.