If `y z+z x+x y=12 ,w h e r ex ,y ,z` are positive values, find the greatest value of `x y zdot`

Explanation

Given, `yz+zx+xy=12` (constant), the value of `(yz)(zx)(xy)` is greatest when `yz=zx=xy`
Hence, `n=3` and `k=12`
Hence, greatest value of `(yz)(zx)(xy)` is `((12)/(3))^(3)` i.e.64.
`:.` Greatest value of `x^(2)y^(2)z^(2)` is 64.
Thus, greatest value of xyz is 8.
Aliter
Given `yz+zx+xy=12`, the greatest value of (yz)(zx)(xy) is greatest when
`yz=zx=xy=c " " [" say "]`
Since, `yz+zx+xy=12`
`:. c+c+c=12`
`implies 3c=12` r `c=4`
`:. yz=zx=xy=4`
Hence, greatest value of (yz)(zx)(xy) is `4*4*4`
i.e. greatest value of `x^(2)y^(2)z^(2)` is 64.
Hence, greatest value of xyz is 8.