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

If you believed $x=y=3$ is probably the answer, then you could shift the problem to be centered there: that often makes things clearer.
So define $x' = x - 3$ and $y' = y-3$. Our problem is now that $x' + y' = 0$ and we want to minimize $(x'+3)^2 + (y'+3)^2$. If we expand that, we get
$$ x'^2 + 6x' + 9 + y'^2 + 6y' + 9 = x'^2 + y'+2 + 6(x' + y') + 18 = x'^2 + y'^2 + 18$$
This is obviously minimized when $x' = y' = 0$, as desired.
The above would look essentially the same if you solved $x+y=6$ for $y$ to rewrite the entire problem in terms of $x$ before doing the above.
