Question

1,cho x+y=2.CMR:\(x^{2011}+y^{2011}\le x^{2012}+y^{2012}\)

Answer 1

Áp dụng BĐT Cauchy:

\(2011.x^{2012}+1\ge2012.x^{2011}\) ; \(2011y^{2012}+1\ge2012x^{2011}\)

\(\Rightarrow2011\left(x^{2012}+y^{2012}\right)\ge2011\left(x^{2011}+y^{2011}\right)+x^{2011}+y^{2011}-2\)

Mặt khác \(x^{2011}+2010\ge2011x\) ; \(y^{2011}+2010\ge2011y\)

\(\Rightarrow x^{2011}+y^{2011}\ge2011\left(x+y\right)-2010.2=2\)

\(\Rightarrow2011\left(x^{2012}+y^{2012}\right)\ge2011\left(x^{2011}+y^{2011}\right)\)

\(\Rightarrow x^{2012}+y^{2012}\ge x^{2011}+y^{2011}\)

Dấu "=" xảy ra khi \(x=y=1\)