Calculus
posted by bobby .
Find two numbers whose sum is 10 for which the sum of their squares is a minimum.

two numbers : x and 10x
S = x^2 + (10x)^2
dS/dx = 2x  2(10x)
= 0 for max/min
x = 5
so the numbers are both 5 
x and (10x)
s = x^2 + (10x)^2
s = x^2 + 100 20x + x^2
s = 2 x^2  20 x + 100
s/2 = x^2 10 x + 100 we can minimize half the sum easier than the whole sum
That is a parabola and you could find the vertex but since you said this was "calculus" we will take the derivative and set to zero.
0 = 2 x  10
x = 5
10x = 5 
Interesting the answer is halfway between.
Exploring that
Say a sum of two numbers is s
We want to minimize the sum of squares of x^2 and (sx)^2
sum = 2 x^2 2sx
d sum/dx = 0 = 4 x 2s
x = s/2
so it works for any old sum, not just 10
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