You function is
f = x^2 + y^2 - (2 x^2 + 2 y^2 - x)^2;
Then call function Maximize
Maximize[f, {x, y}](*{27/64,{x->3/4,y->0}}*)
And here is plot of you function
You function is
f = x^2 + y^2 - (2 x^2 + 2 y^2 - x)^2;
Then call function Maximize
Maximize[f, {x, y}](*{27/64,{x->3/4,y->0}}*)
And here is plot of you function