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

Explanation

Let ` P= xy^(3)`
Givne, ` x+y = 60 rArr x = 60 - y`
`:. P= (60-y)y^(3) rArr P= 60 y^(3) - y^(4)`
`rArr (dP)/(dy) = 180 y^(2) - 4y^(3)`
and ` (d^(2)P)/(dy^(2)) = 360 y - 12y^(2)`
For maximum value, ` (dP)/(dy) = 0`
` rArr 180y^(2) - 4y^(3)=0`
` rArr 4y^(2) (45-y)=0`
` rArr y= 0 or y= 45`
but `y ne 0,` so y = 45
at ` y = 45, ((d^(2)P)/(dy^(2)))_(y=45) = 360 xx 45 - 12 xx (45)^(2)`
`= 16200 - 24300=- 8100 lt 0`
` rArr ` P has local maximum value at y = 45.
` :. x = 60 - 45 = 15`
So, the numbers are 15 and 45.