Question

Tìm giá trị của x để

D=\(\frac{x^2-2x+2014}{x^2}\), x khác 0 đạt GTNN

 

Answer 1

\(D=\frac{x^2-2x+2014}{x^2}\)

\(D=\frac{x^2}{x^2}-\frac{2x}{x^2}+\frac{2014}{x^2}\)

\(D=1-\frac{2}{x}+\frac{2014}{x^2}\)

\(D=2014\cdot\frac{1}{x^2}-2\cdot\frac{1}{x}+1\)

Đặt \(\frac{1}{x}=a\)

\(D=2014a^2-2a+1\)

\(D=2014\left(a^2-a\cdot\frac{1}{1007}+\frac{1}{2014}\right)\)

\(D=2014\left(a^2-2\cdot a\cdot\frac{1}{2014}+\frac{1}{2014^2}+\frac{2013}{2014^2}\right)\)

\(D=2014\left[\left(a-\frac{1}{2014}\right)^2+\frac{2013}{2014^2}\right]\)

\(D=2014\left(a-\frac{1}{2014}\right)^2+\frac{2013}{2014}\ge\frac{2013}{2014}\forall x\)

Dấu "=" xảy ra \(\Leftrightarrow a=\frac{1}{2014}\Leftrightarrow\frac{1}{x}=\frac{1}{2014}\Leftrightarrow x=2014\)

Vậy....