Let $x,y \in R$ such that $|x+y| + |x-y| = 2$ 
Let $x,y \in R$ such that $|x+y| + |x-y| = 2$, then find maximum values of $x^2 - 6x + y^2$ and $x^2 + y^2 + 10y$.

How do I go about solving this question? Is it possible to find all real values of $x$ and $y$ from the first equation? Please help.

Hint:
If $f(x,y)=|x+y|+|x-y|-2,$
observe that  $f(x,y)=f(-x,y)=f(x,-y)=f(-x,-y)$
WLOG $x+y, x-y\ge0;$
$$x=1, -x\le y\le x, y^2\le x^2=1$$
Now,
$$x^2+y^2+10y=1+(y+5)^2-25$$
$-1\le y\le1\iff5-1\le5+y\le5+1\implies4^2\le(y+5)^2\le6^2$
