two numbers : x and 10-x
S = x^2 + (10-x)^2
dS/dx = 2x - 2(10-x)
= 0 for max/min
x = 5
so the numbers are both 5
x and (10-x)
s = x^2 + (10-x)^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
10-x = 5
Interesting the answer is halfway between.
Say a sum of two numbers is s
We want to minimize the sum of squares of x^2 and (s-x)^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
Math - Minimizing Sum: Find two positive numbers whose sum is 100 and the sum of...
math - Find two positive numbers whose sum is 50 such that the sum of their ...
Math - Find two numbers whose difference is 16, and the sum of whose squares a ...
calculus - Find two positive numbers whose product is 100 and whose sum is a ...
Pre-Calculus - If the sum of two numbers is 1 and the sum of the cubes of the ...
math - find two numbers whose product is N^2 and whose sum is a minimum
math - Two non-negative numbers are chosen such that their sum is 30. Find the ...
Math - Find two numbers whose ratio is 4 to 3 and are such that the sum of their...
math - find two numbers whose sum is 25 and the sum of whose reciprocals is one ...
math - The sum of two numbers is 10.The sum of their squares is 52.Find the ...