Question

Tim GTNN của A = x^2+y^2/x^2+2xy+y^2

Answer 1

\(A=\dfrac{x^2+y^2}{x^2+2xy+y^2}\)

\(2A=\dfrac{2x^2+2y^2}{\left(x+y\right)^2}\)

\(2A=\dfrac{x^2+2xy+y^2+x^2-2xy+y^2}{\left(x+y\right)^2}\)

\(2A=1+\dfrac{\left(x-y\right)^2}{\left(x+y\right)^2}\)

Do : \(\dfrac{\left(x-y\right)^2}{\left(x+y\right)^2}\) ≥ 0 ∀xy

⇒ \(2A=1+\dfrac{\left(x-y\right)^2}{\left(x+y\right)^2}\) ≥ 1

⇔ \(A\) ≥ \(\dfrac{1}{2}\)

⇒ A_Min = \(\dfrac{1}{2}\) ⇔ x = y