Divide the number 20 into two parts (not necessarily integers) in a way that makes the product of one part with the square of the other as large as possible

use calculus methods

13 1/3 and  6 2/3.

Step-by-step explanation:

Let the 2 parts be x and (20 - x).

So x^2(20 - x) must be a maximum.

y = x^2(20 - x)

y = 20x^2 - x^3

Finding the derivative):

y' =  40x - 3x^2 = 0   for  maxm/minm value.

x( 40 - 3x) = 0

40 - 3x = 0

3x = 40

x = 40/3.

This is a maximum because the second derivative y" =  40 - 6x = 40 - 6(40/3)) is negative.

So the 2 numbers are 13 1/3 and  6 2/3.

the answer is 0052

0 0 5 2

5 0 2 6 3 5

− 0

2 6

− 0

2 6 3

− 2 5 0

1 3 5

− 1 0 0

3 5