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Chapter 13: Problem 51

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.

Short Answer

Expert verified
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 by step solution

01

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.
02

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.
03

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.

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