Find the maximum value of xyz when `(x)/(1)+(y^2)/(4)+(z^3)/(27)=1`, where `x,y,zgt 0`.

Explanation

we have `(x)/(1) + (y^(2))/(4) + (z^(3))/(27) = 1`
Using weighted means, we have
`(x + (1)/(2).(y^(2))/(2)+(1)/(3).(z^(3))/(9))/(1 + (1)/(2) + (1)/(3)) ge [x((y^(2))/(2))^(1//2) ((z^(3))/(9))^(1//3)]^((1)/(1 + (1)/(2) + (1)/(3)))`
`implies (1)/((11)/(6)) ge [x((y^(2))/(2))^(1//2) ((z^(3))/(9))^(1//3)]^((6)/(11)`
`implies xyz ((1)/(2))^(1//2) ((1)/(9))^(1//3) le ((6)/(11))^((11)/(6))`
`implies xyz le ((6)/(11))^((11)/(6)) .^(2)sqrt(2). .^(3)sqrt(9)`
So, maximum value of xyz is `((6)/(11))^((11)/(6)) .^(2)sqrt(3), .^(3)sqrt(9)`.