Optimizing Sum of Squares with Two Positive Numbers

[Agent_J]

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?

 

[matt grime]

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.

 

[mathman]

You don't need calculus.

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