Question

[Nguyễn Thị Sao Mai]

Cho x,y\(>0\) thỏa mãn x+y=1. Chứng minh: \(\left(x+\frac{1}{x}\right)^2+\left(y+\frac{1}{y}\right)^2\ge\frac{25}{2}\)

Answer 1

[Thắng Nguyễn]

Áp dụng BĐT AM-GM ta có: 

\(A=\left(x+\frac{1}{x}\right)^2+\left(y+\frac{1}{y}\right)^2\ge\frac{\left(x+\frac{1}{x}+y+\frac{1}{y}\right)^2}{2}\)

\(=\frac{\left(x+y+\frac{1}{x}+\frac{1}{y}\right)^2}{2}=\frac{\left(x+y+\frac{x+y}{xy}\right)^2}{2}\)

Lại có: \(1=x+y\ge2\sqrt{xy}\Rightarrow1\ge4xy\Rightarrow\frac{1}{xy}\ge4\)

Khi đó \(A\ge\frac{\left(1+\frac{1}{xy}\right)^2}{2}=\frac{\left(1+4\right)^2}{2}=\frac{5^2}{2}=\frac{25}{2}\)

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