$f(x, y) = x^4 + y^4 + 2 x^{2}y^{2} − 2 x^2 + 2 y^2 + 1$
i- What are the domain and range of the function? Is the domain closed? open? bounded? Give brief justifications to your answers.
ii- Draw the level curve passing through $(\sqrt{2}, 0)$.
iii- Find the tangent line (or lines, if there is more than one) to the level curve from part (ii) at $(0,0)$.
iv- Find the maximum value of the function $g(x, y) = y^2 − x^2$ when it is restricted to the curve in part (ii).
Hint: You might find polar coordinates useful for some parts of this question.