# Given $x, y in mathbb{R}^+$ and $x^3 + y^3 = x-y$. Prove $x^2 + 4y^2 < 1$

Mathematics Asked by user9026 on September 14, 2020

For $$x, y in mathbb{R}^+$$ , I am given that $$x^3 + y^3 = x-y$$. I have to prove that $$x^2 + 4y^2 < 1$$. Now I have $$y + y^3 = x(1-x^2)$$. Since $$y + y^3 > 0$$ and $$x > 0$$, we have $$1 – x^2 > 0$$. Which means that $$0 < x < 1$$. Also, since $$x^3 + y^3 > 0$$, we have $$x – y > 0$$. So, $$x > y$$. So, we get $$0 < y < x < 1$$. Using AM-GM inequality, I also get

$$(x^3 + y^3 + 1) geqslant 3xy$$

What else can be dome here ?

Thanks

We need to prove that $$x^2+4y^2 or since $$x-y>0$$, $$y(5y^2-4xy+x^2)>0,$$ which is obvious because $$5y^2-4xy+x^2=y^2+(2y-x)^2>0.$$

Correct answer by Michael Rozenberg on September 14, 2020

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