Find two numbers whose sum is 100 and whose sum of squares is a minimum. How do you know it is a minimum?

Question

HELP me please!

Answer 1

try 36 plus 64 equals 100
and the sum of 6 plus 8 (the square roots of 36 and 64) is 14

not sure if this makes sense
but all I could figure out, hope it helps

Answer 2

It depends if the numbers need to be different or not.

If they do, then the numbers you are looking for are 49 and 51. They add up to 100, and their squares add up to 5002 (49x49 + 51x51 = 2401 + 2601 = 5002).

If they don't, then you can get slightly better by using 50 and 50. That also adds up to 100, but the squares are only 5000 (50x50 + 50x50 = 2500 + 2500 = 5000).

We know that these are the minimums because squares are exponential functions - higher squares get bigger much faster, so a big number squared plus a small number squared will always be higher than two medium numbers with the same sum squared. Because 50 and 50 (or 49 and 51) are the two sets of numbers that add up to 100, and have no very high numbers, they must be the minimum.

I'm assuming they need to be Whole numbers, and not decimals.

Answer 3

Let one of the numbers be x.
Then the other one is 100 - x.
The sum of the squares is:
x^2 + (100 - x)^2.
= 2x^2 + 10000 - 200x
= 2(x^2 - 100x + 5000)
= 2 [(x - 50)^2 + 2500]
Since the squared term cannot be less than zero, the minimum value occurs when x = 50.
This means x and 100 - x are both 50, and the sum of the squares is 2*2500 = 5000.