Local extrema

Definition

Local extrema are points on a graph where the function reaches a local maximum or minimum within a given interval. These points are identified by analyzing the first derivative of the function.

5 Must Know Facts For Your Next Test

1. A local maximum is a point where the function changes direction from increasing to decreasing.
2. A local minimum is a point where the function changes direction from decreasing to increasing.
3. Local extrema occur where the first derivative of the function is zero or undefined.
4. The second derivative test can help determine whether a critical point is a local maximum, local minimum, or neither.
5. Local extrema are important in understanding the overall shape and behavior of a graph.

Review Questions

• What is the difference between a local maximum and a local minimum?
• How do you find potential local extrema using the first derivative?
• What role does the second derivative play in identifying local extrema?

Related terms

Critical Point: A point on the graph where the first derivative is zero or undefined.

First Derivative Test: A method used to determine whether a critical point is a local maximum, local minimum, or neither based on sign changes in the first derivative.

Second Derivative Test: A method used to classify critical points by evaluating the concavity of the function at those points using its second derivative.