Question

Let $f(x, y) = |xy| + x$ for all $(x, y) \in \mathbb{R}^2$. Determine the existence of the partial derivative of $f$ with respect to $x$ exists:

• A. at $(0,0)$ but not at $(0,1)$.
• B. at $(0,1)$ but not at $(0,0)$.
• C. at both $(0,0)$ and $(0,1)$.
• D. neither at $(0,0)$ nor at $(0,1)$.

Answer 1

Answer: (A)

at $(0,0)$ but not at $(0,1)$.

Answer 2

The correct option is (A): at $(0,0)$ but not at $(0,1)$.