What is the smallest value of $x^2+y^2$ when $x+y=6$? If $ x+y=6 $ then what is the smallest possible value for $x^2+y^2$?
Please show me the working to show where I am going wrong!
Cheers

We have 
$$2(x^2+y^2)=(x+y)^2 +(x-y)^2=36+(x-y)^2.$$ 
But $36+(x-y)^2$ is smallest when $x=y$. Thus the minimum value of $2(x^2+y^2)$ is $36$. 
