Question

Cho x,y,z là các số dương thỏa mãn x+y+z=2. Tìm GTNN của

P=\(\sqrt{4x^2+\frac{1}{x^2}}+\sqrt{4y^2+\frac{1}{y^2}}+\sqrt{4z^2+\frac{1}{z^2}}\)

Answer 1

\(P=\sqrt{\left(2x\right)^2+\left(\frac{1}{x}\right)^2}+\sqrt{\left(2y\right)^2+\left(\frac{1}{y}\right)^2}+\sqrt{\left(2z\right)^2+\left(\frac{1}{z}\right)^2}\)

\(P\ge\sqrt{\left(2x+2y+2z\right)^2+\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right)^2}\)

\(P\ge\sqrt{4\left(x+y+z\right)^2+\frac{81}{\left(x+y+z\right)^2}}=\frac{\sqrt{145}}{2}\)

Dấu "=" xảy ra khi \(x=y=z=\frac{2}{3}\)