\(A=x^2+20y^2+8xy-4y+2015\)
\(=\left(x^2+8xy+16y^2\right)+\left(4y^2-4y+1\right)+2014\)
\(=\left(x+4y\right)^2+\left(2y-1\right)^2+2014\ge2014\forall x\)
Dấu "=" xảy ra khi:
\(\hept{\begin{cases}x+4y=0\\2y-1=0\end{cases}\Rightarrow}\hept{\begin{cases}x=-2\\y=\frac{1}{2}\end{cases}}\)
Vậy GTNN của A là 2014 khi \(x=-2,y=\frac{1}{2}\)
\(B=\frac{x^2-2x+2016}{x^2}\)
\(=\frac{2016x^2-2.x.2016+2016^2}{2016x^2}\)
\(=\frac{\left(x^2-2.x.2016+2016^2\right)+2015x^2}{2016x^2}\)
\(=\frac{\left(x-2016\right)^2+2015x^2}{2016x^2}=\frac{\left(x-2016\right)^2}{2016x^2}+\frac{2015}{2016}\ge\frac{2015}{2016}\forall x\)
Dấu "=" xảy ra khi: \(x-2016=0\Rightarrow x=2016\)
Vậy GTNN của B là \(\frac{2015}{2016}\)khi x = 2016
