Problem 51 - College Algebra and Calculus: An Applied Approach

Question

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. A saddle point always occurs at a critical point.

Step-by-Step Solution

Answer

The statement is true. A saddle point always occurs at a critical point, which is illustrated by the example of function \( f(x,y) = x^2 - y^2 \), which has a saddle point at the critical point (0, 0).

Step 1: Understand the Terms

To begin, it's important to understand the terms involved. A critical point for a function occurs when the function's derivative is 0 or does not exist. A saddle point, on the other hand, is a point on the surface of the graph of a function where the concave changes. Although it is a type of critical point, not all critical points are saddle points.

Step 2: Evaluate the Statement

Given the definitions, it can be deduced that the statement 'A saddle point always occurs at a critical point' is true. As previously established, a saddle point is a type of critical point, so it must occur at some point where the derivative is zero or does not exist.

Step 3: Provide an Example

For instance, consider the function \( f(x,y) = x^2 - y^2 \). This function has a critical point at (0, 0) because the partial derivatives are both 0 at this point. Additionally, the Hessian determinant, which is defined as \( D = f_{xx}f_{yy} - (f_{xy})^2 \), is negative at this point, indicating that it is a saddle point. Hence, the statement is true, and a saddle point indeed occurs at a critical point in this example.