I was messing around with the function $$f_k(x)=\frac{1}{k+1}\prod_{n=1}^{k} (x-n)^{\frac{1}{2n-1}}$$ because it has a funny looking graph. It was easy to determine that the zeroes are at $$x=1, 2, 3, ..., k$$ But I also wanted to find the maxima and minima. After working at it for a while, I determined that $$f_k'(x)=\frac{1}{k+1}\sum_{m=1}^{k}\bigg( \frac{1}{(x-m)(2m-1)}\prod_{n=1}^{k} (x-n)^{\frac{1}{2n-1}} \bigg)$$ Now I'm stuck. How the heck do I find the zeroes of this?
Just one more thing...
I also want to find a closed-form formula for the $y$-intercept in terms of $k$. This means that I would have to find a closed-form for $$f_k(x)=\frac{(-1)^{k}}{k+1}\prod_{n=1}^{k} n^{\frac{1}{2n-1}}$$ I also have no clue how to do this. Maybe it's really easy? I don't know that much about how to evaluate products like that.
All help is appreciated!