Split 20 into two non negative numbers x; y such that the product of x and y^2 is a maximum

Degrees of freedom rules tell that the number of unknowns should be equal to the number of equations involved in the system of equations. The numerical or algebraic expressions of the two given statements are represented by the equations:

(1) 20 = x+y

(2) P = x y^2

where x and y are the unknowns and P is the product and the maximum.

When finding for the maximum, we take the first derivative of the function and equate to zero. we can substitute x by the equation in (1). Hence,

P = xy^2 = (20-y) y^2 = 20y^2 - y^3

P' = 40 y - 3y^2 = 0

getting y,

40y = 3y^2

40 = 3y

y = 40/3

x = 56/3

The two numbers then are fractions 40/3 and 56/3.