Create a function f(x,y) that have at least 1 minimum, 1 maximum and 1 saddle point.
**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 \( f(x, y) = x^3 - 3x + y^2 \) 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.

Explore how to create a function with a minimum, maximum, and saddle point using multivariable calculus concepts. Understand the Hessian determinant's role in confirming critical point classifications with a practical example.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation