Question

Giúp em với ạ, em cảm ơn

Answer 1

\(T=\left(\dfrac{x}{2}+\dfrac{y}{2}-\dfrac{x\sqrt{y}+y\sqrt{x}}{x+y}+\dfrac{3}{4}\right)\left(4040+\dfrac{x}{y}+\dfrac{y}{x}\right)\)

\(T\ge\left(\dfrac{x}{2}+\dfrac{y}{2}-\dfrac{x\sqrt{y}+y\sqrt{x}}{2\sqrt{xy}}+\dfrac{3}{4}\right)\left(4040+\dfrac{x}{y}+\dfrac{y}{x}\right)\)

\(T\ge\left(\dfrac{x}{2}+\dfrac{y}{2}-\dfrac{\sqrt{x}}{2}-\dfrac{\sqrt{y}}{2}+\dfrac{3}{4}\right)\left(4040+2\sqrt{\dfrac{xy}{xy}}\right)\)

\(T\ge\left[\dfrac{1}{2}\left(\sqrt{x}-\dfrac{1}{2}\right)^2+\dfrac{1}{2}\left(\sqrt{y}-\dfrac{1}{2}\right)^2+\dfrac{1}{2}\right].4042\)

\(T\ge\dfrac{4042}{2}=2021\)

Dấu "=" xảy ra khi \(x=y=\dfrac{1}{4}\)