In general, how do you know that a function increases or decreases? Take an easier example, I know that the function g(x)=x increases over its whole domain. How could I prove that to you?
So .
Since we need to show that , I guess the next step would be to consider the numerator and denominator separately. Consider the denominator first:
is always positive since are by definition positive over the interval .
It remains to show that the numerator is negative over the whole interval, i.e., . In other words we need to show that .
I am not sure how to actually show this, since it is so obvious to me by looking at the graph. I thought about trying some kind of induction but we are not operating over N so it would be hard. Any further guidance you can give me?
Here is my complete proof, does this work:
So .
Since we need to show that , I guess the next step would be to consider the numerator and denominator separately. Consider the denominator first:
is always positive since are by definition positive over the interval .
It remains to show that the numerator is negative over the whole interval, i.e., . In other words we need to show that .
Note: , so we have:
Now is clearly true (or do I need to show something more) so it follows that the numerator of is negative . Hence is decreasing over the whole interval.
That is up to you. You can use the fact that for any , since .
Edit:
If you want to use what Plato wrote, you would want to show that is a strictly increasing function on . That is obvious since for any positive . Hence, since , , and the rest of what Plato wrote follows.