Global minimum

Definition

The global minimum of a function is the lowest point over its entire domain. It represents the smallest value that the function can take.

5 Must Know Facts For Your Next Test

1. The global minimum is found by evaluating the function at critical points and endpoints.
2. For polynomial functions, the global minimum may occur at infinity if the degree of the polynomial is even.
3. A function can have only one global minimum, but it may have multiple local minima.
4. To identify a global minimum, you must compare all local minima and boundary values.
5. The second derivative test can help determine whether a critical point is a local minimum or maximum, which aids in finding the global minimum.

Review Questions

• How do you find the global minimum of a polynomial function?
• What is the difference between a local minimum and a global minimum?
• Why is it important to evaluate endpoints when looking for a global minimum?

Related terms

Local Minimum:

A point where a function takes on a value lower than at any other nearby points.

Critical Point: A point where the derivative of a function is zero or undefined.

Second Derivative Test: $$\text{A method to determine if a critical point is a local maximum or minimum by analyzing the sign of the second derivative}.$$