Kí hiệu: $x\sqrt{1-y^2}+y\sqrt{1-z^2}+z\sqrt{1-x^2}=\frac{3}{2}$ $(*)$Theo bất đẳng thức $AM - GM$ ta có:
$x\sqrt{1-y^2}+y\sqrt{1-z^2}+z\sqrt{1-x^2}\leq \frac{x^2+1-y^2}{2}+\frac{y^2+1-z^2}{2}+\frac{z^2+1-x^2}{2}=\frac{3}{2}$
Do đó: $(*)\Leftrightarrow \begin{cases}x=\sqrt{1-y^2} \\ y=\sqrt{1-z^2} \\ z=\sqrt{1-x^2} \end{cases}\Leftrightarrow x^2=y^2=z^2=\frac{1}{2}\Rightarrow M=\frac{3}{2}.$