The given function is \( f(x, y) = x^3 - y^3 + 8xy \).

This is a multivariable function, meaning it involves both \(x\) and \(y\). Depending on what you want to do with this function, there are various ways to analyze it, such as finding partial derivatives, critical points, or evaluating it for specific values of \(x\) and \(y\).

Here are some key steps you might be interested in:

1. **Partial derivatives**:
   - Partial derivative with respect to \(x\): \( \frac{\partial f}{\partial x} \)
   - Partial derivative with respect to \(y\): \( \frac{\partial f}{\partial y} \)

2. **Critical points**: By setting the partial derivatives equal to zero, you can solve for critical points.

3. **Second partial derivatives**: To further analyze the behavior around critical points, you can compute second partial derivatives and use the **Hessian matrix**.

4. **Optimization**: You can look for local maxima, minima, or saddle points.

Would you like to explore any of these steps in detail, or do you have a specific question related to this function?

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Here are some related questions:
1. What are the partial derivatives of \(f(x, y)\) with respect to \(x\) and \(y\)?
2. How do you find critical points for \( f(x, y) = x^3 - y^3 + 8xy \)?
3. How can you determine if a critical point is a maximum, minimum, or saddle point?
4. What is the total differential of \(f(x, y)\)?
5. Can you graph the function \( f(x, y) \) to visualize its behavior?

**Tip**: When working with multivariable functions, analyzing the behavior at critical points is essential for understanding the function's geometry.

Explore the function f(x, y) = x^3 - y^3 + 8xy by finding its partial derivatives, critical points, and evaluating behavior at critical points using the Hessian matrix. Suitable for college-level calculus students seeking a deeper understanding of multivariable functions.

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