Question

Find two numbers adding to 20 such that the sum of their squares is as small as possible.

Answer 1

x+y = 20
minimize x^2+y^2

Answer 2

y = 20-x
f(x) = x^2 + (20-x)^2 = x^2 + 400 - 40x + x^2

Answer 3

differentiate:
f'(x) = 2x -40 + 2x = 0,
solve for x. x = 10

Answer 4

test if it's a min: f''(x) = 4 > 0, so x=10 is a min.

Answer 5

y = 20-x=20-10=10

Answer 6

Thanks a ton I can see where I mess up

Answer 7

np, i thought i was messing up too somewhere.. i kept getting 0 for y' for some reason

Answer 8

me too

Answer 9

bet anything you make them equal right?

Answer 10

yes

Answer 11

calc will work but you can argue by symmetry, since you cannot tell the difference between x and y

Answer 12

\[x^2+y^2\]
\[x+y=20\] switch x and y and get
\[y^2+x^2\]
\[y+x=20\] meaning you cannot tell the difference between x and y

Answer 13

just thought i would mention it