Local Minimum

Definition

A local minimum is a point on a graph where the function value is smaller than the function values at all nearby points. It represents a point where the function reaches a minimum within a local region, even if it may not be the absolute minimum of the entire function.

5 Must Know Facts For Your Next Test

1. Local minima are important in understanding the behavior and shape of a function, as they represent points where the function is decreasing and then begins to increase again.
2. The first derivative test can be used to identify local minima, where the derivative changes from negative to positive at the critical point.
3. Local minima may not be the absolute minimum of the function, as there could be lower points elsewhere in the function's domain.
4. The second derivative test can also be used to identify local minima, where the second derivative is positive at the critical point.
5. Understanding local minima is crucial in optimization problems, where the goal is to find the minimum or maximum value of a function.

Review Questions

• Explain how local minima are related to the behavior of a function's graph.
  • Local minima represent points on a function's graph where the function value is smaller than the function values at all nearby points. These points indicate a local minimum, where the function is decreasing and then begins to increase again. Local minima are important in understanding the overall shape and behavior of a function, as they provide information about the function's critical points and regions of concavity.
• Describe how the first and second derivative tests can be used to identify local minima.
  • The first derivative test can be used to identify local minima by looking for points where the derivative changes from negative to positive. At these critical points, the function is transitioning from decreasing to increasing, indicating a local minimum. The second derivative test can also be used, where a positive second derivative at a critical point signifies a local minimum. The second derivative test provides additional information about the concavity of the function around the critical point, further confirming the presence of a local minimum.
• Explain the significance of local minima in optimization problems and how they differ from absolute minima.
  • Local minima are crucial in optimization problems, where the goal is to find the minimum or maximum value of a function. Local minima represent points where the function reaches a minimum within a local region, even if they may not be the absolute minimum of the entire function. While local minima are important in understanding the function's behavior, the absolute minimum is the overall smallest value the function takes on over its entire domain. In optimization problems, both local and absolute minima are important considerations, as the goal may be to find the global, or absolute, minimum, or to identify all local minima that may represent viable solutions.

Related terms

Critical Point: A critical point is a point on a function where the derivative is equal to zero or undefined. Local minima and maxima are types of critical points.

Absolute Minimum: The absolute minimum of a function is the smallest value the function takes on over its entire domain, regardless of local variations.

Concavity: Concavity refers to the curvature of a function, which can be used to identify local minima and maxima.