Question

Cho x,y dương thỏa mãn \(\sqrt{xy}\left(x-y\right)=x+y\)

Tìm min C=x+y

Answer 1

Từ pt => x>y>0

pt<=>\(\left(x+y\right)^2=xy\left(x-y\right)^2\Leftrightarrow\left(x+y\right)^2=\left(\left(x+y\right)^2-4xy\right)xy\)

Đặt x+y=a, xy=b (a,b>0)

pttt \(a^2=\left(a^2-4b\right)b\Leftrightarrow a^2-a^2b+4b^2=0\Leftrightarrow\left(4b^2-a^2b+\frac{a}{16}^4\right)+a^2-\frac{a^4}{16}=0\)

\(\Leftrightarrow\left(2b-\frac{a}{4}^2\right)=\frac{a}{16}^4-a^2\)

Do VT >= 0 => VP>=o\(\Leftrightarrow a^2\ge16\Leftrightarrow a\ge4\)do a>0

Answer 2

mình sắp tốt nghiệp cấp 3 rồi :(