Find two non negative numbers whose sum is 30

Question

such that the product of one number times the square of the other number is a minimum.? 

Answer 1

Let the non negitive numbers are x and y

Thier sum is 30

Then it's algebraic expression is x + y = 30

The product of one number times square of the other number is a minimum.

f (x, y ) = xy ^2

we need to create xy ^2 as a function of either x or y .

y  = 30 - x

x (30 - x )^2  = x (900 + x ^2  - 60x )

y = 900x + x ^3 - 60x ^2

derivative the function.

f (x ) = x ^3 - 60x ^2 + 900x

f '(x) = 3x ^2 - 120x + 900

To find minimum ,where the derivative equals to zero.

 3(x ^2 - 40x + 300) = 0

x ^2 - 30x - 10x + 300 = 0

 x (x - 30) - 10(x - 30)] = 0

(x - 30)(x - 10) = 0

x - 30 = 0 and x - 10 = 0

x = 30 and x = 10

If x  = 30 then  y = 30 - x

y = 30 - 30 = 0

If x = 10 then y = 30 -10 = 20

The non-trivial answer is when x = 10, y = 20.

The non negitive numbers are x = 10 , y = 20