Question

Cho x, y, z dương thoả mãn

\(\sqrt{x}+\sqrt{y}+\sqrt{z}=3\)
tìm GTNN của \(M=\sqrt{x+\frac{1}{x}}+\sqrt{y+\frac{1}{y}}+\sqrt{z+\frac{1}{z}}\)

Answer 1

Đặt \(\left(\sqrt{x};\sqrt{y};\sqrt{z}\right)=\left(a;b;c\right)\Rightarrow a+b+c=3\)

\(M=\sqrt{a^2+\frac{1}{a^2}}+\sqrt{b^2+\frac{1}{b^2}}+\sqrt{c^2+\frac{1}{c^2}}\)

\(M\ge\sqrt{\left(a+b+c\right)^2+\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)^2}\)

\(M\ge\sqrt{\left(a+b+c\right)^2+\frac{81}{\left(a+b+c\right)^2}}\)

\(M\ge\sqrt{2\sqrt{\frac{81\left(a+b+c\right)^2}{\left(a+b+c\right)^2}}}=3\sqrt{2}\)

\(M_{min}=3\sqrt{2}\) khi \(a=b=c=1\)