Minimize $x^2y^2z^2$ if $x^2+y^2+z^2=4$
Well I want to minimize it using classical inequalities; because I have easily maximized it with $AM\ge GM$ as follows: $$x^2+y^2+z^2\ge3\sqrt[3]{x^2y^2 z^2}$$ $$\iff 4 \ge3\sqrt[3]{x^2y^2 z^2} \iff \frac{4}{3}\ge \sqrt[3]{x^2 y^2z^2}$$ $$\iff \frac{64}{27}\ge x^2y^2z^2$$ but I don't know how to find the minimum using classical inequalities. Is there a way?
Any help would be appreciated!
Let $z=0$ and $y=z=\sqrt2$.
Thus, $$x^2y^2z^2=0$$ and since $$x^2y^2z^2\geq0$$ for any $x$, $y$ and $z$ such that $x^2+y^2+z^2=4$, we got a minimal value.
Id est, $$\min_{x^2+y^2+z^2=4}x^2y^2z^2=0.$$