Local extrema

Definition

Local extrema are points on a graph where a function reaches a local maximum or minimum value. These points represent the highest or lowest values within a specific interval of the function.

5 Must Know Facts For Your Next Test

1. Local maxima occur where the function changes from increasing to decreasing, while local minima occur where it changes from decreasing to increasing.
2. At local extrema, the derivative of the function is either zero or undefined.
3. Second derivative test can be used to determine whether a critical point is a local maximum, minimum, or neither.
4. Local extrema are not necessarily global extrema; they are only extreme within their immediate vicinity.
5. The first derivative test involves analyzing the sign changes of the first derivative around critical points to identify local extrema.

Review Questions

• How do you identify whether a critical point is a local maximum or minimum using the second derivative test?
• What is the significance of the first derivative being zero at local extrema?
• Explain how you would use the first derivative test to find local extrema on a given function.

Related terms

Critical Point: A point on a graph where the first derivative is zero or undefined, potentially indicating a local extremum.

First Derivative Test: A method for determining whether a critical point is a local maximum or minimum by analyzing the sign changes of the first derivative.

Second Derivative Test: A method for determining concavity at critical points to classify them as local maxima, minima, or saddle points based on the sign of the second derivative.