SOLUTION: Find two nonnegative numbers whose sum is 12 such that their product is an absolute maximum.

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Question 1191565: Find two nonnegative numbers whose sum is 12 such that their
product is an absolute maximum.
Found 3 solutions by Alan3354, Edwin McCravy, greenestamps:
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
Find two nonnegative numbers whose sum is 12 such that their product is an absolute maximum.
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6 and 6.
It's always equal numbers
In this case, the product is 36.
x*x = 36 is the max and x+x = 12
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Any other combination is less than 36
(x-1)*(x+1) = x^2 - 1 = 35
(x-2)*(x+2) = x^2 - 4 = 32
(x-k)*(x+k) = x^2 - k^2 = 36 - k^2

Answer by Edwin McCravy(20055) About Me  (Show Source):
You can put this solution on YOUR website!
Instead of doing yours for you, I'll do one exactly like it so you can
use it as a model to do yours by.  The one I'll do for you is:
Find two nonnegative numbers whose sum is 10 such that their
product is an absolute maximum.
Let x be one of the nonnegative numbers.
Then, since their sum is 10, the other nonnegative number is 10-x

Let y = their product = x(10-x) = 10x-x2 = -x2+10x

We graph y = -x2+10x where a=-1, b=10, c=0


 
The vertex is the highest point. So we find the coordinates of the vertex,
using the vertex formula:

1.  x-coordinate of vertex = -b%2F%282a%29%7D%7D%7B%7B%7B%22%22=%22%22-%2810%29%2F%282%28-1%29%29%22%22=%22%225

2.  y-coordinate, substitute x-coordinate in equation for parabola,
y=-x%5E2%2B10x%22%22=%22%22-%285%29%5E2%2B10%285%29=-25%2B50=25

So the vertex is (5,25) which occurs when x=5

So one of the nonnegative numbers is x = 5
The other nonnegative number is 10-x = 10-5 = 5

So the product of two nonnegative numbers whose sum is 10 is a maximum 
of 25 when the two nonnegative numbers are both 5.

Now do yours the same way.

Edwin

Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


Let the two numbers whose sum is 12 be 6+x and 6-x; their product is then 36-x^2.

x^2 is always 0 or positive, so the maximum value of 36-x^2 is when x^2 is 0, which means x is 0.

The two positive numbers with a sum of 12 for which the product of the two numbers is maximum are then 6+0=6 and 6-0=6.

ANSWER: 6 and 6