Question

Tìm các số dương x, y, z biết: \(x^2+y^2+z^2+\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}=6\)

 

Answer 1

(x^2-2+1/x^2 ) +( y^2-2+1/y^2) +(z^2-2+1/z^2) =0

=> (x-1/x)^2 +(y-1/y)^2+(z-1/z)^2=0

suy ra x-1/x=0 

          y-1/y=0

         z-1/z=0

.....

Answer 2

Ta có: \(x^2+\frac{1}{x^2}\ge2\sqrt{x^2.\frac{1}{x^2}}=2\)

\(y^2+\frac{1}{y^2}\ge2\sqrt{y^2.\frac{1}{y^2}}=2\)

\(z^2+\frac{1}{z^2}\ge2\sqrt{x^2.\frac{1}{z^2}}=2\)

\(\Rightarrow VT\ge6\)

Dấu "=" khi \(\orbr{\begin{cases}x=y=z=1\\x=y=z=-1\end{cases}}\)