minimize the sum of the squares

[lalala123]

Hi again, (got the other one now thanks!! :lol:

Last one:

2 Positive numbers have sum n. What is the smallest possible value for the sum of their squares?

so i put the 2 numbers as x and y.
n=x+y x>0 y>0

so I need to minimize
x^2+y^2

i know its quite similar to the one i asked previously but Im very confused since I am not given any numbers to work with. THANKS

 

[royhaas]

If it's only two numbers, just substitute y=n-x and minimize. This problem generalizes to the case of k arbitrary numbers with a fixed sum.

 

[lalala123]

still very confused!!!

hi,
im still quite confused

so i do that so x+y=n
y=n-x
so n=x+n-x
but that doesnt seem right that goes to n-n=x-x which is 0=0????
Please help

 

[pka]

You finish; find x to minimize S.
\(\displaystyle \L
\begin{array}{l}
S = x^2 + y^2 \quad \& \quad y = n - x \\
S = x^2 + \left( {n - x} \right)^2 \\
S' = 2x - 2\left( {n - x} \right) = 4x - 2n \\
\end{array}\)