S8.3.7.4. The sum of two positive numbers is 16.

In summary, the smallest possible value for the sum of squares of two positive numbers that add up to 16 is 128. This can be found by completing the square and setting the derivative to 0, resulting in x=8, with a corresponding sum of squares of 128.
  • #1
karush
Gold Member
MHB
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3.7.4. The sum of two positive numbers is 16.

What is the smallest possible value of the sum of their squares?

$x+y=16\implies y=16-x$
Then
$x^2+(16-x)^2=2 x^2 - 32x + 256$

So far
... Hopefully
 
Last edited:
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  • #2
keep going ... ATQ
 
  • #3
[3.7.4. The sum of two positive numbers is 16.
What is the smallest possible value of the sum of their squares?

$x+y=16\implies y=16-x$
Then
$z=x^2+(16-x)^2=2 x^2 - 32x + 256$
Then
$z'=4x-32$
So
$z'(x)=0\quad x=8$
Check
8^2+8^2=128
 
  • #4
karush said:
[3.7.4. The sum of two positive numbers is 16.
What is the smallest possible value of the sum of their squares?

$x+y=16\implies y=16-x$
Then
$z=x^2+(16-x)^2=2 x^2 - 32x + 256$
Then
$z'=4x-32$
So
$z'(x)=0\quad x=8$
Check
8^2+8^2=128
Question: Is that the smallest value for x?

-Dan
 
  • #5
topsquark said:
Question: Is that the smallest value for x?

-Dan

Question: Is that what was asked in the question? I thought they were asking for the smallest sum of squares...
 
  • #6
Prove It said:
Question: Is that what was asked in the question? I thought they were asking for the smallest sum of squares...
Okay. Yes, you are right. I was trying to get karush to prove that this was the minimum answer in order to round out his solution. He never proved that.

-Dan
 
  • #7
Rather than use something as "advanced" as setting the derivative to 0, I would "complete the square". \(\displaystyle 2x^2- 32x+ 256= 2(x^2- 16x+ 128)= 2(x^2- 16x+ 64+ 64)= 2((x- 8)^2+ 64)\).

That will be smallest when x= 8 and then the value will be 2(64)= 128.
 
Last edited by a moderator:

FAQ: S8.3.7.4. The sum of two positive numbers is 16.

1. What are the two positive numbers?

The two positive numbers cannot be determined without additional information. All we know is that their sum is 16.

2. Can the two numbers be fractions or decimals?

Yes, the two numbers can be fractions or decimals as long as their sum is still equal to 16.

3. Is 16 the only possible sum of two positive numbers?

No, there are infinite combinations of two positive numbers that can add up to 16. For example, 8 and 8, 10 and 6, 12 and 4, etc.

4. Can the two numbers be equal?

Yes, the two numbers can be equal if they are both 8. However, this is just one of many possible combinations.

5. Can the two numbers be negative?

No, the question states that the two numbers are positive, so they cannot be negative.

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