tthandb

Ta có: $x^{2}+y^{3}\geq x^{3}+y^{4}\Rightarrow x^{2}+y^{3}+y^{2}\geq x^{3}+y^{2}+y^{4}$

Áp dụng BĐT AM-GM ta có $y^{4}+y^{2}\geq 2y^{3}$

Do đó: $x^{2}+y^{3}+y^{2}\geq x^{3}+2y^{3}\Rightarrow x^{3}+y^{3}\leq x^{2}+y^{2}$ (1)

Áp dụng BĐT Cauchy-Swcharz, ta có :

$\left ( x^{2}+y^{2} \right )^{2}\leq \left [ \left ( \sqrt{x} \right )^{2}+\left ( \sqrt{y} \right )^{2} \right ]\left [ \left ( \sqrt{x^{3}} \right )^{2}+\left ( \sqrt{y^{3}} \right )^{2} \right ]=\left ( x+y \right )\left ( x^{3}+y^{3} \right )$

           $\leq \left ( x+y \right )\left ( x^{2}+y^{2} \right )\Rightarrow x^{2}+y^{2}\leq x+y$  (2)

 Mặt khác $\left ( x+y \right )^{2}\leq 2\left ( x^{2}+y^{2} \right )\leq 2\left ( x+y \right )\Rightarrow x+y\leq 2$ (3)

Từ (1),(2),(3) suy ra đpcm

P/s: by thuan192