Optimization Problem! PLEASE HELP

Question

Find two non-negative numbers whose sum is 21 and whose product is a minimum.

Answer 1

Symmeatra,

 

Let x & y be the two non-negative numbers.  So each can zero or positive.

Minimize the product function(x,y) = xy

 

But they are constrained by the condition that x + y = 21 so you can build a table:

 

x          y           xy

---      ----        -----

0         21            0

1         20           20

2         19           38

...

9         12          108

10       11          110

11       10          110

12         9          108

...

20         1            20

21         0              0

 

So their product is minimized at 0 whenever either x or y is zero while the other number is 21.  Each row in the table satisfies the sum = (x + y) =21 so 0 <= x <= 21 and 0 <= y <= 21 is the domain of f(x,y) = xy.

 

Or you can use Calculus, take partial derivates of f(x,y) wrt x and y, set them to 0 and get x =0 and y=0, but not simultaneously because the sum would not be 21 then. So calculate the other variable from (x + y) = 21.  And you get the same solution as above, but quicker.

 

Answer 2

So, if you represent one number as x, the other number can be represented as 21-x. Their product can be represented as:

F(x)=21x-x^2

 

Now if this is a calculus problem, take the derivative, set it equal to 0 and solve.

 

If this is a lower level class, you have to realize that this is a parabola and the maximum or minimum value will always be on the axis of symmetry. The formula for the axis of symmetry is x = -b/2a.

 

By either method, this problem has a maximum value when x=10.5

So to answer what is the minimum value is either a mistake or an unanswerable question since there is no such thing as the smallest positive number. 

 

I just reread the problem carefully and since the numbers are non negative, 0 can be an answer and therefore 0 can be the product. So your numbers are 0 and 21.

 

Hope this helped.