Question

cho x,y>0 thỏa mãn x^2+y^3>=x^3+y^4. cmr x^3+y^3=<2

Answer 1

\(x^2+y^3+y^2\ge x^3+y^4+y^2\ge x^3+2y^3\Rightarrow x^2+y^2\ge x^3+y^3\)

Lại có \(\left(x^2+y^2\right)^2=\left(\sqrt{x}.\sqrt{x^3}+\sqrt{y}\sqrt{y^3}\right)^2\le\left(x+y\right)\left(x^3+y^3\right)\)

\(\Rightarrow\left(x^2+y^2\right)^2\le\left(x+y\right)\left(x^2+y^2\right)\Rightarrow x^2+y^2\le x+y\)

\(\Rightarrow\left(x^2+y^2\right)^2\le\left(x+y\right)^2\le2\left(x^2+y^2\right)\)

\(\Rightarrow x^2+y^2\le2\Rightarrow x^3+y^3\le2\)

Dấu "=" xảy ra khi \(x=y=1\)