Inequality $\frac{x}{x^{10}+1}+\frac{y}{y^{10}+1}+\frac{z}{z^{10}+1}\leq \frac{3}{2}$

Question

I try to prove this :

Let $x,y,z>0$ such that $xyz=1$ then we have : $$\frac{x}{x^{10}+1}+\frac{y}{y^{10}+1}+\frac{z}{z^{10}+1}\leq \frac{3}{2}$$

You can follow my proof here.It works too but I have some doubt for the end so my real question is :

How to prove that the maximum of the function below is $1.5$ : $$f(x)=\frac{2x^9}{x^{10}+1}+\frac{x^2}{x^{20}+1}$$

Thanks for your interest .