$($From Hardcover Book, Marsden$/$Tromba$,$ Vector Calculus, $6$th ed$.,$ Exercises for Section $3\mathrm{.}4,$ # $17)$

Find the maximum and minimum values attained by $f(x,y,z)=xyzf(x,y,z)=xyz$ on the unit ball $x^2{y}^2{z}^{2}1x^2{y}^2{z}^{2}1.$