{x -> 0.244984},
{x -> 1.10815},
{x -> 3.64288},
{x -> 6.57833},
{x -> 9.62956},
{x -> 12.7223},
{x -> 15.8336},
{x -> 18.9547},
{x -> 22.0815},
{x -> 25.2119},
{x -> 28.3448},
{x -> 31.4794}
I just dumped your function into Mathematica and numerically solved for the first 10 or so zeros of it's derivative. That's about all you can do here.
Hi,
It always helps to sketch a graph. By peaks, I assume you mean relative maxima. In the attachment, the original function is shown in black and your derivative is in red. It looks from the graph that the maxima are all approximately on the line y = x. Maybe you or someone else could investigate this.