Ta có \(A=\frac{x^2-2x+2011}{x^2}\)

\(=\frac{x^2}{x^2}-\frac{2x}{x^2}+\frac{2011}{x^2}\)

\(=1-\frac{2}{x}+\frac{2011}{x^2}\)

Đặt \(\frac{1}{x}=y\)ta có:

\(A=1-2y+2011y^2\)

\(A=2011y^2-2y+1\)

\(A=2011\left(y^2-\frac{2}{2011}y+\frac{2}{2011}\right)\)

\(=2011\left(y^2-2\times y\times\frac{1}{2011}+\frac{1}{2011^2}-\frac{1}{2011^2}+\frac{1}{2011}\right)\)

\(=2011\left(\left(y-\frac{1}{2011}\right)^2\right)+\frac{2010}{2011^2}\)

\(=2011\left(y-\frac{1}{2011}\right)^2+\frac{2010}{2011}\)

Vì (y-\(\frac{1}{2011}\))\(^2\)>=0

\(\Rightarrow2011\left(y-\frac{1}{2011}\right)^2+\frac{2010}{2011}\)

Hay \(A>=\frac{2010}{2011}\)

quyet

bài này ta có thể giải theo 2 cách 

ta có A = \(\frac{x^2-2x+2011}{x^2}\)

= \(\frac{x^2}{x^2}\)- \(\frac{2x}{x^2}\)+ \(\frac{2011}{x^2}\)

= 1 - \(\frac{2}{x}\)+ \(\frac{2011}{x^2}\)

đặt \(\frac{1}{x}\)= y ta có 

A= 1- 2y + 2011y^2 

cách 1 : 

A = 2011y^2 - 2y + 1 

= 2011 ( y^2 - \(\frac{2}{2011}y\)+ \(\frac{1}{2011}\)) 

= 2011( y^2 - 2.y.\(\frac{1}{2011}\)+ \(\frac{1}{2011^2}\)- \(\frac{1}{2011^2}\) + \(\frac{1}{2011}\)) 

= 2011 \(\left(\left(y-\frac{1}{2011}\right)^2\right)+\frac{2010}{2011^2}\)

= 2011\(\left(y-\frac{1}{2011}\right)^2\)+ \(\frac{2010}{2011}\)

vì ( y - \(\frac{1}{2011}\)) ²>=0 

=> 2011\(\left(y-\frac{1}{2011}\right)^2\)+ \(\frac{2010}{2011}\)> = \(\frac{2010}{2011}\)

hay A >=\(\frac{2010}{2011}\)

cách 2  

A = 2011y^2 - 2y + 1 

= ( \(\sqrt{2011y^2}\)) - 2 . \(\sqrt{2011y}\). \(\frac{1}{\sqrt{2011}}\)+ \(\frac{1}{2011}\)+ \(\frac{2010}{2011}\)

= \(\left(\sqrt{2011y}-\frac{1}{\sqrt{2011}}\right)^2\)+ \(\frac{2010}{2011}\)

vì \(\left(\sqrt{2011y}-\frac{1}{\sqrt{2011}}\right)^2\)> =0 

nên \(\left(\sqrt{2011y}-\frac{1}{\sqrt{2011}}\right)^2\)+ \(\frac{2010}{2011}\)>= \(\frac{2010}{2011}\)

hay A >= \(\frac{2010}{2011}\)

Ta có :

 \(B=\frac{x^2-2x+2011}{x^2}\)

\(B=\frac{x^2}{x^2}-\frac{2x}{x^2}+\frac{2011}{x^2}\)

\(B=1-\frac{2}{x}+\frac{2011}{x^2}\)

\(B=\left(\frac{\sqrt{2011}^2}{x^2}-\frac{2}{x}+\frac{1}{2011}\right)+\frac{2010}{2011}\)

\(B=\left(\frac{\sqrt{2011}}{x}-\frac{1}{\sqrt{2011}}\right)^2+\frac{2010}{2011}\)

Mà : \(\left(\frac{\sqrt{2011}}{x}-\frac{1}{\sqrt{2011}}\right)^2\ge0\forall x\)

\(\Rightarrow B\ge\frac{2010}{2011}\)

Dấu "=" xảy ra khi :

\(\frac{\sqrt{2011}}{x}-\frac{1}{\sqrt{2011}}=0\)

\(\Leftrightarrow x=2\sqrt{2011}\)

Vậy \(MinB=\frac{2010}{2011}\Leftrightarrow x=2\sqrt{2011}\)

\(\frac{2011^3+11^3}{2011^3+2000^3}=\frac{\left(2011+11\right)\left(2011^2+11^2-11.2011\right)}{\left(2011+200\right)\left(2011^2+2000^2-2000.2011\right)}\)

Cần chứng minh \(2011^2+11^2-2011.11=2011^2+2000^2-2000.2011\)

Điều này không khó.

\(B=1-\frac{2}{x}+\frac{2011}{x^2}=2011t^2-2t+1\text{ (với }t=\frac{1}{x}\text{)}\)

->Gộp hằng đẳng thức....

\(A=\left|\left(x+1\right)^2+\left(y-2\right)^2\right|-\left(x+y-1\right)^2+2xy\)

\(=\left(x+1\right)^2+\left(y-2\right)^2-\left(x^2+y^2-2x-2y+2xy+1\right)+2xy\)

\(=4x-2y+4\)

thay số.Lưu ý: \(y=16^{503}=\left(2^4\right)^{503}=2^{2012}\)