Maximum of $\frac{xy^2z}{x^4 - y^4 + z^4}$ with positive variables

Question

What's the maximum value of the equation $$ \frac{xy^2z}{x^4 - y^4 + z^4} $$

with $x, y, z > 0$?

I have tried to divide it with $x + y + z$ but nothing came out of there.

Answer 1

It is unbounded. To see this, notice that the values of $x^4+z^4$ and $xz$ are always positive and independent of $y$. Thus, it is sufficient to consider the function $$f(y) = \frac{ay^2}{b-y^4},$$

which clearly approaches $\pm \infty$ as $y \rightarrow b^{1/4} = (x^4+z^4)^{1/4}$.

Answer 2

Just take $x = y = 1$ and $z = 1/t$. Then your expression becomes $t^3$ and thus is unbounded.

Answer 3

This answer is a slight variation of the answer given by J.-E. Pin

Let $x = y = 1$. Then your expression becomes $\Large \frac{1}{z^3}$. For positive values of $z$, this expression becomes arbitrarily large as $z$ approaches $0$.