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\(A=\frac{x^2-2x+2007}{2007x^2}=\frac{2006}{2007^2}+\frac{x^2-4014x+2007^2}{2007^2x^2}=\frac{2006}{2007^2}+\frac{\left(x-2007\right)^2}{2007^2x^2}\ge\frac{2006}{2007^2}\)
Dấu ''='' xảy ra \(\Leftrightarrow\) x = 2007
\(A=\frac{2007x^2-2x.2007+2007^2}{2007x^2}\)
\(=\frac{x^2-2x.2007+2007^2}{2007x^2}+\frac{2006x^2}{2007x^2}\)
\(=\frac{\left(x-2007\right)^2}{2007x^2}+\frac{2006}{2007}\ge\frac{2006}{2007}\)
A min =\(\frac{2006}{2007}\)khi \(x-2007=0\) hay \(x=2007\)
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a, \(A=\left(\frac{4}{2x+1}+\frac{4x-3}{\left(x^2+1\right)\left(2x+1\right)}\right)\frac{x^2+1}{x^2+2}\)
\(=\left(\frac{4\left(x^2+1\right)}{\left(2x+1\right)\left(x^2+1\right)}+\frac{4x-3}{\left(x^2+1\right)\left(2x+1\right)}\right)\frac{x^2+1}{x^2+2}\)
\(=\left(\frac{4x^2+4+4x-3}{\left(x^2+1\right)\left(2x+1\right)}\right)\frac{x^2+1}{x^2+2}\)
\(=\frac{\left(2x+1\right)^2}{\left(x^2+1\right)\left(2x+1\right)}\frac{x^2+1}{x^2+2}=\frac{2x+1}{x^2+2}\)
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a)\(\frac{x^2+4}{x^2}+\frac{4}{x+1}\left(\frac{1}{x}+1\right)\)
\(=\frac{x^2+4}{x^2}+\frac{4}{x+1}.\frac{x+1}{x}\)
\(=\frac{x^2+4}{x^2}+\frac{4}{x}\)
\(=\frac{x^2+4x+4}{x^2}\)
\(\left(\frac{x+2}{x}\right)^2\)
=>phép chia = 1 với mọi x # 0 và x#-1
b)Cm tương tự
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Ta có:
\(2A=\frac{2x^2+4x+6}{\left(x+2\right)^2}=\frac{\left(x^2+4x+4\right)+x^2+2}{\left(x+2\right)^2}=1+\frac{x^2+2}{\left(x+2\right)^2}\)
Đặt \(B=\frac{x^2+2}{\left(x+2\right)^2}\) và \(y=x+2\Leftrightarrow x=y-2\)
Vì \(A\) đạt giá trị nhỏ nhất \(\Leftrightarrow\) \(B\) nhỏ nhất nên ta có:
\(B=\frac{\left(y-2\right)^2+2}{y^2}=\frac{y^2-4y+4+2}{y^2}=\frac{y^2-4y+6}{y^2}=1-\frac{4}{y}+\frac{6}{y^2}\)
\(B=\frac{1}{3}+\frac{2}{3}-\frac{4}{y}+\frac{6}{y^2}=\frac{1}{3}+\left(\sqrt{\frac{2}{3}}\right)^2-2.\sqrt{\frac{2}{3}.}\frac{\sqrt{6}}{y}+\left(\frac{\sqrt{6}}{y}\right)^2\)
\(B=\frac{1}{3}+\left[\left(\sqrt{\frac{2}{3}}\right)-\frac{\sqrt{6}}{y}\right]^2\ge\frac{1}{3}\) với mọi \(y\)
Do đó:
\(2A=1+\frac{1}{3}+\left[\left(\sqrt{\frac{2}{3}}\right)-\frac{\sqrt{6}}{y}\right]^2\)
\(2A=\frac{4}{3}+\left[\left(\sqrt{\frac{2}{3}}\right)-\frac{\sqrt{6}}{y}\right]^2\ge\frac{4}{3}\) với mọi \(y\)
\(\Rightarrow\) \(A\ge\frac{2}{3}\)
Dấu \(''=''\) xảy ra \(\Leftrightarrow\left[\left(\sqrt{\frac{2}{3}}\right)-\frac{\sqrt{6}}{y}\right]^2=0\)
\(\Leftrightarrow\sqrt{\frac{2}{3}}-\frac{\sqrt{6}}{y}=0\)
\(\Leftrightarrow y=3\)
\(\Leftrightarrow x=1\)
Vậy \(Min\) \(A=\frac{2}{3}\) \(\Leftrightarrow\) \(x=1\)
\(A=\frac{2007x^2-2x.2007+2007^2}{2007x^2}=\frac{x^2-2x.2007+2007^2}{2007x^2}+\frac{2006x^2}{2007x^2}\)
\(=\frac{\left(x-2007\right)^2}{2007x^2}+\frac{2006}{2007}\ge\frac{2006}{2007}\)
A min =\(\frac{2006}{2007}\)khi \(x-2007=0\)
\(\Leftrightarrow x=2007\)
\(A=\frac{2007x^2-2x.2007+2007^2}{2007x^2}\)
\(A=\frac{x^2-2x.2007-2007^2}{2007x^2}+\frac{2006x^2}{2007x^2}\)
\(A=\frac{\left(x-2007\right)^2}{2007x^2}+\frac{2006}{2007}\ge\frac{2006}{2007}\)
\(\Rightarrow Amin=\frac{2006}{2007}\)khi \(x-2007=0\)
\(\Rightarrow x=2007\)