Question

Cho x,y > 0 thỏa mãn x + 4/y ≤ 2 tìm gtln của P = 2xy / x² + 3xy + y²

Answer 1

\(2\ge\dfrac{4}{y}+x\ge2\sqrt{\dfrac{4x}{y}}\Rightarrow\dfrac{y}{x}\ge4\)

\(\dfrac{2}{P}=\dfrac{x^2+3xy+y^2}{xy}=\dfrac{x}{y}+\dfrac{y}{x}+3=\left(\dfrac{x}{y}+\dfrac{y}{16x}\right)+\dfrac{15}{16}.\dfrac{y}{x}+3\ge2\sqrt{\dfrac{xy}{16xy}}+\dfrac{15}{16}.4+3=\dfrac{29}{4}\)

\(\Rightarrow P\le\dfrac{8}{29}\)

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