Question

So sánh : 

\(\sqrt{2004}+\sqrt{2006}\) và \(2\sqrt{2005}\)

Answer 1

Giả sử : \(\sqrt{2004}+\sqrt{2006}< 2\sqrt{2005}\)

\(\Leftrightarrow2004+2006+2\sqrt{2004.2006}< 4.2005\)

\(\Leftrightarrow\sqrt{2004.2006}< 2005\Leftrightarrow2004.2006< 2005^2\)

\(\Leftrightarrow\left(2005-1\right)\left(2005+1\right)< 2005^2\)

\(\Leftrightarrow2005^2-1< 2005^2\) . BĐT đúng

Vậy \(\sqrt{2004}+\sqrt{2006}< 2\sqrt{2005}\)

Answer 2

Giả sử : \(\sqrt{2004}+\sqrt{2006}< 2\sqrt{2005}\) 

\(\Leftrightarrow2004+2006+2\sqrt{2004.2006}< 4.2005\)

\(\Leftrightarrow\sqrt{2004.2006}< 2005\Leftrightarrow2004.2006< 2005^2\)

\(\Leftrightarrow\left(2005-1\right)\left(2005+1\right)< 2005^2\)

\(\Leftrightarrow2005^2-1< 2005^2.\) BĐT đúng

Vậy \(\sqrt{2004}+\sqrt{2006}< 2\sqrt{2005}\)