Consider the function

$$f(x, y)=x^{2}+y^{2}-\frac{1}{2} x^{4}-b x^{2} y^{2}-\frac{1}{2} y^{4}$$

where $b$ is a positive constant.

Find the critical points of $f(x, y)$, assuming $b \neq 1$. Determine the type of each critical point and sketch contours of constant $f(x, y)$ in the two cases (i) $b<1$ and (ii) $b>1$.

For $b=1$ describe the subset of the $(x, y)$ plane on which $f(x, y)$ attains its maximum value.