Question

[Incursion_03]

\(\hept{\begin{cases}x+\frac{1}{x}=y^2+1\left(1\right)\\y+\frac{1}{y}=z^2+1\left(2\right)\\z+\frac{1}{z}=x^2+1\left(3\right)\end{cases}}\left(ĐKXĐ:x;y;z\ne0\right)\)

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

              \(\Rightarrow x=\frac{x^2+1}{y^2+1}>0\)

Tương tự \(y=\frac{y^2+1}{z^2+1}>0\)

                \(z=\frac{z^2+1}{x^2+1}>0\)

Áp dụng bđt cô-si có \(x+\frac{1}{x}\ge2\) 

\(\Rightarrow y^2+1\ge2\)

\(\Rightarrow y\ge1\)

Tương tự \(x;z\ge1\)

ta có \(xyz=\frac{\left(x^2+1\right)\left(y^2+1\right)\left(z^2+1\right)}{\left(x^2+1\right)\left(y^2+1\right)\left(z^2+1\right)}=1\)

Mà \(x;y;z\ge1\Rightarrow xyz\ge1\)

Do đó dấu "=" khi x = y = z = 1 

Yey =)))

Answer 1

[Mo Anime]

thanks

Answer 2

[Dương Phạm]

Thanks gì bạn Mo ???