Q29Multiple correct3 Marks22 Dec 2024

Consider the function $f: \mathbb{R}^2 \to \mathbb{R}$ defined as follows: $$f(x, y) = \begin{cases} (x - y)^3 \sin\left(\frac{1}{x - y}\right) & \text{for } x \neq y \\ 0 & \text{for } x = y \end{cases}$$ Choose all the correct statements from the following.

A

The function f is continuous at $(0, 0)$.

B

There is a direction along which the directional derivative of f does not exist at $(0, 0)$.

C

The equation of the tangent hyperplane to the graph of f at the point $(0, 0)$ is given by z = 0.

D

The vector $(1, 1, -1)$ is orthogonal to the tangent hyperplane to the graph of f at $(0, 0)$.