Question

Let \( f : \mathbb{R}^2 \to \mathbb{R} \) be given by

\[ f(x, y) = \begin{cases} \frac{x^2 y (x - y)}{x^2 + y^2}, & (x, y) \neq (0, 0) \\ 0, & (x, y) = (0, 0) \end{cases} \]

Then

\[ \frac{\partial}{\partial x} \left( \frac{\partial}{\partial y} f \right) - \frac{\partial}{\partial y} \left( \frac{\partial}{\partial x} f \right) \text{ at the point } (0, 0) \text{ is ..........} \]

Hint:

When computing mixed partial derivatives, always check for continuity and differentiability at the point of interest, especially when dealing with piecewise functions.

Answer 1

Correct Answer: 1

Step 1: Compute partial derivatives.
We need to compute the mixed partial derivatives of \( f \). First, we compute the partial derivatives with respect to \( x \) and \( y \) for \( f(x, y) \). We calculate these at \( (0, 0) \).

Step 2: Apply limits at \( (0, 0) \).
By applying the limits as \( (x, y) \to (0, 0) \), we find that the mixed partial derivatives at \( (0, 0) \) simplify to 0.

Step 3: Conclusion.
Thus, the correct value is \( \boxed{0} \).