Prove the inequality

*Proof.* Define a function . Then, since for we have

Since and is increasing on (since its derivative is positive) we have

For the inequality on the left (which is much more subtle), we want to show

So, we consider the function

We know

and

Now, if we can show that is decreasing on the whole interval then we will be done (since this would mean on the whole interval since if it were less than somewhere then it would have to increase to get back to on the right end of the interval).

To show is decreasing on the whole interval we will show that its derivative is negative. To that end, define a function

(This is the numerator in the expression we got for . Since we know the denominator of that expression is always positive, we are going to show this is always negative to conclude is always negative.) Now, and

for . Thus, is negative, and so is decreasing. Since we then have is negative on the whole interval. Therefore, on the interval. Hence, is decreasing. Hence, we indeed have

for all