Create a function f(x,y) that have at least 1 minimum, 1 maximum and 1 saddle point(have to have 3 of them).
**Use the Hessian determinant to confirm your answer. You can also try to sketch its graph using 3D graphing tool.

The 3D plot of the function \( f(x, y) = x^3 - 3x + y^3 - 3y \) confirms the nature of the critical points:

- **Local Minimum** at \((1, 1)\).
- **Local Maximum** at \((-1, -1)\).
- **Saddle Points** at \((1, -1)\) and \((-1, 1)\).

This behavior matches our analysis using the Hessian determinant.

### Follow-Up Questions:

1. How does changing the coefficients of \( x \) and \( y \) affect the location of the critical points?
2. Can you find another function with the same type of critical points but located at different positions?
3. How would adding a linear term, like \( x + y \), to the function affect its critical points?
4. What would happen to the Hessian determinant if the function had higher-degree terms?
5. How do critical points relate to the overall shape of the function?

**Tip:** Analyzing the second derivatives can provide deep insights into the geometry of a function around critical points!

Explore how to create a multivariable function f(x, y) with at least one minimum, one maximum, and one saddle point. Understand the critical points at (1, 1), (-1, -1), (1, -1), and (-1, 1) using the Hessian determinant. Discover how changing coefficients in the function affects critical point locations and the function's overall shape.

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