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!