I didn't look at this at length, but you may try expanding it out to:
Now, since perhaps you can simplify it down further.
I took a closer look at your problem and, if I am correct, it has an interesting
max and min. I got as the max and
as the min. It's related to The Golden Ratio.
We can see from differentiating or graphing that the max occurs at
Unless I made a mistake. Check it out. Interesting solution if correct.