Problem

Let $D=\{(x,y):1\le x^2+y^2\le 2,\ x\ge 0\text{ and }y\ge 0\}$ and

$$f(x,y)= \begin{cases} \dfrac{xy}{x^2+y^2}, & x^2+y^2>0,\\ 0, & \text{otherwise}. \end{cases}$$

(a) (5 pts) Determine whether $f$ is continuous at $(0,1)$. Justify your answer.

(b) (5 pts) Find $f_y(0,0)$.

(c) (5 pts) Find $\int_D f(x,y)\,d(x,y)$.

(d) (5 pts) Does $f$ have a relative minimum at $(0,0)$? Justify your answer.