I have the following problem : let $x$ a real positive number such that $x\ge 1$ and $\alpha\ge 0$ a real positive number prove that :
$$f(x)=\frac{x^3}{x+1}+\frac{1}{2}+\frac{x^{\alpha}}{x^{\alpha}+1} -1.5(3)^{\frac{-2}{3}}(x^3+2)^{\frac{2}{3}}\geq 0$$
I try some classical inequalities like Hölder , but it fails .
Thanks a lot .
you will get $$f'(x)=-\frac{x^3}{(x+1)^2}+\frac{3 x^2}{x+1}-\frac{2 x}{3^{2/3} \sqrt[3]{x^2+2}}+\frac{\alpha x^{\alpha -1}}{x^{\alpha }+1}-\frac{\alpha x^{2 \alpha -1}}{\left(x^{\alpha }+1\right)^2}$$ and compute $f(1)$ and Show that $$f'(x)\geq 0$$ for $$x\geq 1$$