Áp dụng bất đẳng thức Mincopski
\(\Rightarrow\sqrt{x^2+1}+\sqrt{y^2+1}+\sqrt{z^2+1}\ge\sqrt{\left(x+y+z\right)^2+9}\)
Chứng minh rằng : \(\sqrt{\left(x+y+z\right)^2+9}\ge\sqrt{6\left(x+y+z\right)}\)
\(\Leftrightarrow\left(x+y+z\right)^2+9\ge6\left(x+y+z\right)\)
\(\Leftrightarrow\frac{\left(x+y+z\right)^2+9}{x+y+z}\ge6\)
\(\Leftrightarrow x+y+z+\frac{9}{x+y+z}\ge6\)
Áp dụng bất đẳng thức Cauchy - Schwarz
\(\Rightarrow x+y+z+\frac{9}{x+y+z}\ge2\sqrt{\frac{9\left(x+y+z\right)}{x+y+z}}=2\sqrt{9}=6\left(đpcm\right)\)
Vậy \(\sqrt{\left(x+y+z\right)^2+9}\ge\sqrt{6\left(x+y+z\right)}\)
Mà \(\sqrt{x^2+1}+\sqrt{y^2+1}+\sqrt{z^2+1}\ge\sqrt{\left(x+y+z\right)^2+9}\)
\(\Rightarrow\sqrt{x^2+1}+\sqrt{y^2+1}+\sqrt{z^2+1}\ge\sqrt{6\left(x+y+z\right)}\left(đpcm\right)\)
Dấu " = " xảy ra khi \(x=y=z=1\)
Chúc bạn học tốt !!!
