Prove $x^3$ is concave down on the interval $(-\infty,0)$ and concave up on the interval $(0,\infty)$.

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Lesson Parent: 
Geometry of Functions IX: Concavity
Problem: 

Prove $x^3$ is concave down on the open interval $(-\infty,0)$ and concave up on the open interval $(0,\infty)$.

SAMPLE SOLUTION

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Solution: 

Before we begin it is best to have a picture in our head of what we are trying to prove:

Concavity of x^3


Recall that a function is concave up on an open interval when its derivative is increasing and concave down when its derivative is decreasing.

First, let's prove that <Sign in to see all the formulas> is concave down on the open interval <Sign in to see all the formulas>. So to prove that $x^3$ is concave down we need to prove that when we go from left to right that the slope is always decreasing. Before we begin, recall that the derivative of $x^3$ is <Sign in to see all the formulas>.

Let's pick two arbitrary values $x_1$ and $x_2$ and assume that <Sign in to see all the formulas>. Now for some simple algebra:

<Sign in to see all the formulas>


Note that $\lt$ changed to $\gt$ in the second step because $x_1$ is a negative number. Similarly,

<Sign in to see all the formulas>


Again, $\lt$ changed to $\gt$ in the second step because $x_2$ is a negative number. Now putting the two results above together we have

<Sign in to see all the formulas>


So as we go from $x_1$ to $x_2$ our derivative decreases (<Sign in to see all the formulas>). Therefore, <Sign in to see all the formulas> is concave down on the open interval <Sign in to see all the formulas>.

Next let's prove that <Sign in to see all the formulas> is concave up on the open interval <Sign in to see all the formulas>. So to prove that $x^3$ is concave up we need to prove that when we go from left to right that the slope is always increasing. Let's pick two arbitrary values $x_1$ and $x_2$ and assume that <Sign in to see all the formulas>. Now for some simple algebra:

<Sign in to see all the formulas>


Similarly,

<Sign in to see all the formulas>


Now putting the two results above together we have

<Sign in to see all the formulas>


So as we go from $x_1$ to $x_2$ our derivative increases (<Sign in to see all the formulas>). Therefore, <Sign in to see all the formulas> is concave up on the open interval <Sign in to see all the formulas>.