(b) (8 points) Show that part (a) becomes wrong for a function of two variables by finding all local maxima and minima of the function: $$ f(x, y) = - (x^2 - 1)^2 - (x^2 y - x - 1)^2. $$ | | | | | |

| 2. Consider $$ f(x, y) = \begin{cases} \dfrac{x y^2}{x^2 + y^2}, & \text{if } (x, y) \neq (0, 0); \\ 0, & \text{if } (x, y) = (0, 0). \end{cases} $$ (a) (3 points) Find the partial derivatives at $(0, 0)$ and the gradient vector $ \nabla f(0, 0) $.