The 3D plot of the function $f(x, y) = x^3 - 3x + y^2$ visually confirms the nature of the critical points:

• The point $(1, 0)$ corresponds to a local minimum.
• The point $(-1, 0)$ corresponds to a saddle point.

This function also exhibits behavior consistent with these classifications based on the Hessian determinant analysis.

Follow-Up Questions:

1. How does the behavior of the Hessian determinant affect the classification of critical points?
2. What would happen if you modified the function to $f(x, y) = x^4 - 4x^2 + y^2$?
3. Can a function have multiple saddle points and still have only one maximum or minimum?
4. How does the shape of the graph change if you modify the coefficients of the function?
5. What is the effect of adding a constant term to $f(x, y)$ on its critical points?

Tip: Understanding the second derivative test helps classify critical points quickly and accurately.