Global minimum

Definition

A global minimum of a function is the lowest point over its entire domain. It is where the function attains its smallest value.

5 Must Know Facts For Your Next Test

1. The global minimum can occur at a critical point or an endpoint of the domain.
2. In polynomial functions, the global minimum can be found by analyzing first and second derivatives.
3. It is essential to differentiate between local and global minima when examining graphs.
4. Global minima are particularly relevant in optimization problems where you seek the least possible value.
5. For even-degree polynomials with positive leading coefficients, there may not be a global minimum as they tend to infinity on both ends.

Review Questions

• What is the difference between a local minimum and a global minimum?
• How can you determine if a critical point is a global minimum?
• Why might an even-degree polynomial with positive leading coefficients not have a global minimum?

Related terms

Critical Point: A point on the graph where the derivative is zero or undefined; potential location for local extrema.

Local Minimum: The lowest point in a particular region of the graph, but not necessarily over the entire domain.

Derivative: A measure of how a function changes as its input changes; used to find slopes of tangents and identify critical points.