Question

TÌm GTLN của B=\(\dfrac{2x^2+8xy+2y^2}{x^2+2xy+y^2}\)

Answer 1

cho x>0,y>0,x+y=2018

Answer 2

\(B=\dfrac{2x^2+8xy+2y^2}{x^2+2xy+y^2}=\dfrac{2x^2+4xy+4xy+2y^2}{x^2+2xy+y^2}\\ =\dfrac{\left(2x^2+4xy+2y^2\right)+4xy}{x^2+2xy+y^2}\\ =\dfrac{2\left(x^2+2xy+y^2\right)}{x^2+2xy+y^2}+\dfrac{4xy}{x^2+2xy+y^2}\\ =2+\dfrac{4xy}{\left(x+y\right)^2}\)

Áp dụng BDT Cô-si : \(4xy\le\left(x+y\right)^2\)

\(\Rightarrow B=2+\dfrac{4xy}{\left(x+y\right)^2}\le2+\dfrac{\left(x+y\right)^2}{\left(x+y\right)^2}\le2+1\le3\)

Dấu \("="\) xảy ra khi: \(x=y\)

Vậy \(B_{\left(Max\right)}=3\) khi \(x=y\)