TL;DR

I want to ask if somebody know a practical example of phenomena that is modelled by the math function cited in the post.

or
$$y=x^{\frac{k}{x}}, \ \ \ \ \ \ \ \ \forall k\in\mathbb{R}$$

thanks,
Ssnow

They start with a fixed quantity and ask for the optimal configuration. Program threads come to mind in parallel programming, where each thread processes a chunk of data. If the chunks are too big, i.e. ie few threads, you might notice that splitting the chunk in two and doubling the thread count improves parallelism and they all terminate sooner.

So then you try smaller and smaller chunks and more threads, but the parallelism vanishes, and it takes longer to process the tiny chunks because each thread has overhead, and more threads mean more memory traffic, which slows things down.

—-

At first I thought this function might relate to the ultraviolet catastrophe in blackbody radiation, since both involve a rise to a peak followed by a decline. However, the comparison does not hold.

In blackbody radiation, the spectrum rises roughly as a power of frequency at low frequencies and then falls off exponentially at high frequencies due to quantum effects introduced from the light quanta introduced by Max Planck.

In contrast, the function x^{2/x} reaches a maximum because increasing the number of terms reduces their individual contribution. Its decay is algebraic rather than exponential.