Question

Cho S = x + y = m-2 , P = xy = 2m+1

tìm gtnn của A = (x^2 + 2)(y^2+2)

Answer 1

Ta có \(A=\left(x^2+2\right)\left(y^2+2\right)=\left(xy\right)^2+2x^2+2y^2+4\)

\(=\left(xy\right)^2+2\left(x+y\right)^2-4xy+4\)\(=\left(2m+1\right)^2+2\left(m-2\right)^2-4\left(2m+1\right)+4\)

\(=4m^2+4m+1+2m^2-8m+8-8m-4+4\)

\(=6m^2-12m+9=6\left(m^2-2m+1\right)+3\)

Ta thấy \(6\left(m-1\right)^2\ge0\Rightarrow6\left(m-1\right)^2+3\ge3\Rightarrow A\ge3\)

Vậy Min A=3 khi m-1=0 hay m=1