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

The Correct Option is A

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