ĐKXĐ: x<>0
Ta có: \(F=\frac{x^2+4x+2015}{2016x^2}\)
\(=\frac{1}{2016}+\frac{4}{2016x}+\frac{2015}{2016x^2}\)
\(=\frac{1}{2016}\left(\frac{2015}{x^2}+\frac{4}{x}+1\right)\)
\(=\frac{2015}{2016}\left(\frac{1}{x^2}+\frac{4}{2015x}+\frac{1}{2015}\right)\)
\(=\frac{2015}{2016}\left(\frac{1}{x^2}+2\cdot\frac{1}{x}\cdot\frac{2}{2015}+\frac{4}{2015^2}-\frac{4}{2015^2}+\frac{1}{2015}\right)\)
\(=\frac{2015}{2016}\left(\frac{1}{x}+\frac{2}{2015}\right)^2+\frac{2015}{2016}\cdot\frac{2011}{2015^2}=\frac{2015}{2016}\left(\frac{1}{x}+\frac{2}{2015}\right)^2+\frac{2011}{2016\cdot2015}\ge\frac{2011}{2016\cdot2015}\forall x\) thỏa mãn ĐKXĐ
Dấu '=' xảy ra khi \(\frac{1}{x}+\frac{2}{2015}=0\)
=>\(\frac{1}{x}=\frac{-2}{2015}\)
=>\(x=-\frac{2015}{2}\)
