Prove $f(x)=\frac{1}{x}$ is concave down on the open interval $(-\infty,0)$ and concave up on the open interval $(0,\infty)$.

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

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

Before we begin, recall that the derivative of $1/x$ is <Sign in to see all the formulas>.

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 $1/x$ is concave down we need to prove that when we go from left to right that the slope is always decreasing. Let's pick two arbitrary values $x_1$ and $x_2$ and assume that <Sign in to see all the formulas>. To prove that it is concave down we need to show that <Sign in to see all the formulas>.

Now for some simple algebra:

<Sign in to see all the formulas>

and

<Sign in to see all the formulas>

Putting these two facts together we see that <Sign in to see all the formulas>. Or more importantly, <Sign in to see all the formulas>. Note that $\lt$ changed to $\gt$ because $x_1$ and $x_2$ are negative numbers. So now for some more algebra:

<Sign in to see all the formulas>

Because we have proven that <Sign in to see all the formulas> we know that $1/x$ is concave down in 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 $1/x$ 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>. To prove this we need to show 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>.