Question

Cho : x + y =1 .Tìm GTNN của :

a) A=x² + y²

b) B=3 - xy

Answer 1

a)Áp dụng BĐT Cauchy-Schwarz ta có:

\(\left(1^2+1^2\right)\left(x^2+y^2\right)\ge\left(x+y\right)^2\)

\(\Rightarrow2\left(x^2+y^2\right)\ge\left(x+y\right)^2\)

\(\Rightarrow2\left(x^2+y^2\right)\ge\left(x+y\right)^2=1^2=1\)

\(\Rightarrow A=x^2+y^2\ge\dfrac{1}{2}\)

Đẳng thức xảy ra khi \(x=y=\dfrac{1}{2}\)

b)Ta có BĐT \(x+y\ge2\sqrt{xy}\Rightarrow\left(x+y\right)^2\ge\left(2\sqrt{xy}\right)^2\)

\(\Leftrightarrow1^2\ge4xy\Leftrightarrow1\ge xy\Leftrightarrow xy\le\dfrac{1}{4}\)

\(\Rightarrow-xy\ge-\dfrac{1}{4}\Rightarrow B=3-xy\ge3-\dfrac{1}{4}=\dfrac{11}{4}\)

Đẳng thức xảy ra khi \(x=y=\dfrac{1}{2}\)