Question

[Lizy]

x,y dương thỏa mãn `xy+4<=2y`. Tìm max `P=(xy)/(x^2 +2y^2)`

Answer 1

[Nguyễn Việt Lâm]

\(2y\ge xy+4\ge2\sqrt{4xy}=4\sqrt{xy}\)

\(\Rightarrow y^2\ge4xy\Rightarrow\dfrac{y}{x}\ge4\)

\(P=\dfrac{xy}{x^2+2y^2}=\dfrac{1}{\dfrac{x}{y}+\dfrac{2y}{x}}=\dfrac{1}{\dfrac{1}{16}\left(\dfrac{16x}{y}+\dfrac{y}{x}\right)+\dfrac{31}{16}.\dfrac{y}{x}}\)

\(\Rightarrow P\le\dfrac{1}{\dfrac{1}{16}.2\sqrt{\dfrac{16xy}{xy}}+\dfrac{31}{16}.4}=\dfrac{4}{33}\)

Dấu "=" xảy ra khi \(\left(x;y\right)=\left(1;4\right)\)