Lấy vế trái trừ vế phải ta có:

\(\frac{2010}{\sqrt{2009}}+\frac{2009}{\sqrt{2010}}-\sqrt{2009}-\sqrt{2010}=\)\(\frac{2010}{\sqrt{2009}}+\frac{2009}{\sqrt{2010}}-\frac{2009}{\sqrt{2009}}-\frac{2010}{\sqrt{2010}}\)=\(\frac{1}{\sqrt{2009}}-\frac{1}{\sqrt{2010}}\) (1)

2009<2010 lên biểu thức (1) >0

đặt a^2010+2009=b

\(\Rightarrow\frac{b+1}{\sqrt{b}}\)

ta có : b+1\(\ge\)2\(\sqrt{b}\)         ( cô - si)

\(\frac{b+1}{\sqrt{b}}\ge2\)

dấu = xảy ra \(\Leftrightarrow b=1\)

\(\Rightarrowđpcm\)
 

+ \(\frac{a}{2009}=\frac{b}{2010}\Leftrightarrow2010a=2009b.\)(1)

+ \(\frac{a+2009}{a-2009}=\frac{b+2010}{b-2010}\Rightarrow\left(a+2009\right)\left(b-2010\right)=\left(a-2009\right)\left(b+2010\right)\)

\(\Rightarrow ab-2010a+2009b-2009.2010=ab+2010a-2009b-2009.2010\)

\(\Leftrightarrow2.2009.b=2.2010.a\Leftrightarrow2010a=2009b\)(2)

Từ (1) và (2) => dpcm

minh lam duoc roi . cach viet phan so ban bam vao o mau vang o cuoi trang .cu di con chuot xuong cuoi trang thi thay 1 o vang , vao xem huong dan la biet ngay ma.

\(B=\frac{2008+2009+2010}{2009+2010+2011}\)

\(=\frac{2008}{2009+2010+2011}+\frac{2009}{2009+2010+2011}+\frac{2010}{2009+2010+2011}\)

\(< \frac{2008}{2009}+\frac{2009}{2010}+\frac{2010}{2011}=A\)

\(B=\frac{2008+2009+2010}{2009+2010+2011}\)

\(=\frac{2008}{2009+2010+2011}=\frac{2009}{2009+2010+2011}=\frac{2010}{2009+2010+2011}\)

\(< A=\frac{2008}{2009}+\frac{2009}{2010}+\frac{2010}{2011}\)

Câu a:

Có dạng tổng quát:\(\frac{1}{\left(k+1\right)\sqrt{k}+k\sqrt{x+1}}=\frac{1}{\sqrt{\left(k+1\right)k}\left(\sqrt{k+1}+\sqrt{k}\right)}=\frac{\sqrt{k+1}-\sqrt{k}}{\sqrt{\left(k+1\right)k}}=\frac{1}{\sqrt{k}}-\frac{1}{\sqrt{k-1}}\)

Áp dụng kết quả trên suy ra câu a