Question

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

Answer 1

4 Ok

Answer 2

\(A=x^2-2xy+6y^2-12x+2y+45\)

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

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

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

Do : \(\left(x-y-6\right)^2\text{≥}0\) ∀\(xy\) ; \(5\left(y-1\right)^2\text{≥}0\text{∀}y\)

⇒ \(\left(x-y-6\right)^2+5\left(y-1\right)^2\text{ ≥}0\)

⇔ \(A=\left(x-y-6\right)^2+5\left(y-1\right)^2+4\text{≥}4\)

⇒ \(A_{Min}=4."="\text{⇔}x=7;y=1\)