Find two positive numbers `x` and `y` such that `x+y=60` and `x y^3` is maximum.

Explanation

`x+y =60`
`rArr " "x=60 -y" "rArr" "xy^(3) =(60 -y)y^(3)`
`"Let "" "f(y) =(60-y) y^(3) " "," "y in (0,60)`
Let maximizing f(y) let us find critical points
`f(y) =3y^(2) (60-y) -y^(3) =0`
`f(y) =y^(2) (180 -4y) =0`
`rArr " "y =45`
`f(45) lt 0 " and " f(45) gt 0.` Hence local maxima at y =45
So `x=15 " and " y=45`