5. Consider the function
$f(x, y) = \begin{cases} \frac{2x^3 + xy + xy^2}{x^2 + 2y^2}, & \text{if } (x, y) \neq (0, 0) \\ 0, & \text{if } (x, y) = (0, 0) \end{cases}$
(a) Find $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ at $(0, 0)$.
(b) Find $\lim_{(x, y) \to (0, 0)} f(x, y)$ if it exists. Justify your answer.
(c) Is $f$ continuous at $(0, 0)$?