Optimizing Sum of Squares with Two Positive Numbers

In summary, the conversation discusses finding two positive numbers whose sum is 100 and the sum of whose squares is a minimum. The conversation then goes on to test different values and concludes that the numbers are likely 50 and 50. However, there is a question about whether the minimum sum is just 50 + 50, or if there are other possible values. It is suggested that calculus may be used to prove that 50 and 50 is the minimum, but it is also mentioned that this can be done without calculus by setting one number as a variable and finding the minimum value of the sum of squares. Ultimately, it is determined that the minimum sum is indeed 50 + 50.
  • #1
Agent_J
13
0
Find 2 positive numbers whose sum is 100 and the sum of whose squares is a minimum. Also, find the minimum sum.

I'm not sure how to write mine, but I just tested 99^2 + 1^2 = 9802
then I tested 50^2 + 50^2 = 5000, so obviously its 50 and 50. Also, I don't understand "find the minimum sum", is it just 50 + 50?
 
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  • #2
who says you are only allowed to consider integers? moreover finding the minimum of x^2+y^2 subject to x+y = 100, and x,y>=0 does not tell you what the minimum is, only where it occurs, when you find the x and y, you are then expected to find the value at that point. as it happpens your guess of 50 and 50 is correct. heuristically, since the question is completely symmetric in x and y an optimum will occur at x=y, however you only believe it to be a minimum at the moment and you need to prove it, say using calculus.
 
  • #3
You don't need calculus.

(x+a)2+(x-a)2=2(x2+a2), which is minimum when a=0, for fixed x (=50).
 

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