https://file.aiquickdraw.com/mathbot/file/17230551750370zv2235i.jpg

Let's analyze the function \( f: \mathbb{R}^2 \to \mathbb{R} \) defined by

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

We are asked to find the value of \( \left( \frac{\partial^2 f}{\partial x \partial y} + \frac{\partial^2 f}{\partial y \partial x} \right) \) at the point \((0, 0)\).

### First, we check the partial derivatives at \((0, 0)\).

#### Partial Derivative \( f_x \) at \( (0,0) \):

\[
f_x(0, 0) = \lim_{h \to 0} \frac{f(h, 0) - f(0, 0)}{h}.
\]

For \( (h, 0) \):

\[
f(h, 0) = \frac{(h \cdot 0)^2}{h+0} = 0.
\]

Therefore,

\[
f_x(0, 0) = \lim_{h \to 0} \frac{0 - 0}{h} = 0.
\]

#### Partial Derivative \( f_y \) at \( (0,0) \):

\[
f_y(0, 0) = \lim_{k \to 0} \frac{f(0, k) - f(0, 0)}{k}.
\]

For \( (0, k) \):

\[
f(0, k) = \frac{(0 \cdot k)^2}{0+k} = 0.
\]

Therefore,

\[
f_y(0, 0) = \lim_{k \to 0} \frac{0 - 0}{k} = 0.
\]

### Second Partial Derivative \( f_{xy} \) at \( (0,0) \):

The second mixed partial derivative \( f_{xy} \) at \( (0, 0) \) is:

\[
f_{xy}(0, 0) = \lim_{h \to 0} \frac{f_y(h, 0) - f_y(0, 0)}{h}.
\]

Given that \( f_y(x, y) = 0 \) for all \( (x, y) \), it follows that \( f_{xy}(0, 0) = 0 \).

### Second Partial Derivative \( f_{yx} \) at \( (0,0) \):

The second mixed partial derivative \( f_{yx} \) at \( (0, 0) is:

\[
f_{yx}(0, 0) = \lim_{k \to 0} \frac{f_x(0, k) - f_x(0, 0)}{k}.
\]

Given that \( f_x(x, y) = 0 \) for all \( (x, y) \), \( f_{yx}(0, 0) = 0 \).

### Conclusion

Since both second mixed partial derivatives \( f_{xy} \) and \( f_{yx} \) at \((0, 0)\) are \(0\),

\[
\left( \frac{\partial^2 f}{\partial x \partial y} + \frac{\partial^2 f}{\partial y \partial x} \right) = 0 + 0 = 0.
\]

Thus, the correct answer is:

\[
\boxed{B}
\]

Would you like more details or have any questions? Here are some related questions:

1. How to compute higher-order partial derivatives for complex functions?
2. What is the importance of mixed partial derivatives in multivariable calculus?
3. Can the partial derivatives at a point provide information about the function’s behavior near that point?
4. What are some common methods to verify the equality of mixed partial derivatives?
5. How does the differentiability of a function relate to the existence of partial derivatives?
6. How can the concept of limits be applied to multivariable functions?
7. What are some applications of partial derivatives in physics and engineering?
8. How can one visualize the partial derivatives of a multivariable function?

**Tip:** When dealing with piecewise functions, always carefully check the points of discontinuity or the specific points given in the problem for partial derivatives.

Explore the analysis of mixed partial derivatives at point (0, 0) for a given multivariable function. This problem involves calculating second-order partial derivatives and verifying their equality using mathematical limits and definitions. Suitable for college-level students studying multivariable calculus.

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