Question

Tìm giá trị nhỏ nhất: A = x² - 2xy + 6y² - 12x + 2y + 45

Answer 1

\(A=x^2-2xy+6y^2-12x+2y+54\)

\(A=x^2-2xy+y^2-12x+12y+36+5y^2-10y+5+4\)

\(A=\left(x-y\right)^2-2.6\left(x-y\right)+36+5\left(y^2-2y+1\right)+4\)

\(A=\left(x-y-6\right)^2+5\left(y-1\right)^2+4\)

Do: \(\left(x-y-6\right)^2\ge0\forall xy\); \(5\left(y-1\right)^2\ge0\forall y\)

\(\Rightarrow\left(x-y-6\right)^2+5\left(y-1\right)^2\ge0\)

\(\Leftrightarrow A=\left(x-y-6\right)^2+5\left(y-1\right)^2+4\ge4\)

\(\Rightarrow A_{Min}=4\)

Dấu "=" xảy ra khi \(x=7;y=1\)