Relative Minimum

Definition

A relative minimum is a point on a graph where the function value is less than or equal to the function values at all nearby points. It represents a local low point on the graph, where the function temporarily decreases before increasing again.

5 Must Know Facts For Your Next Test

1. Relative minima are important in the study of rates of change and the behavior of graphs because they represent local low points where the function is decreasing before increasing again.
2. To identify a relative minimum, you must look for a point where the function value is less than or equal to the function values at all nearby points.
3. Relative minima are often associated with critical points, where the derivative of the function is zero or undefined.
4. The concavity of the graph can also be used to identify relative minima, as they occur at points where the graph changes from concave up to concave down.
5. Relative minima can be used to analyze the overall shape and behavior of a function, including its local and global extrema.

Review Questions

• Explain how relative minima are related to the rates of change of a function.
  • Relative minima represent points on a graph where the function is temporarily decreasing before increasing again. At these points, the rate of change of the function, as measured by the derivative, is zero. This means that the function is neither increasing nor decreasing at its maximum or minimum values, which are important features in analyzing the overall behavior of the graph.
• Describe how the concavity of a graph can be used to identify relative minima.
  • The concavity of a graph, whether it is concave up or concave down, can provide information about the location of relative minima. Relative minima occur at points where the graph changes from concave up to concave down, or vice versa. This change in concavity is often associated with critical points, where the derivative of the function is zero or undefined. By analyzing the concavity of the graph, you can identify the points where the function value is a local minimum.
• Analyze the relationship between relative minima and the overall shape and behavior of a function.
  • Relative minima are important in understanding the overall shape and behavior of a function. They represent local low points on the graph, which can provide insights into the function's global extrema and the intervals where the function is increasing or decreasing. By identifying the relative minima, you can gain a better understanding of the function's critical points, points of inflection, and the regions where the function exhibits different patterns of behavior. This information is crucial in analyzing the rates of change and the overall shape of the graph, which are essential in many mathematical applications.

Related terms

Absolute Minimum:

The absolute minimum of a function is the lowest value the function takes on over its entire domain, in contrast to a relative minimum which is a local low point.

Critical Point: A critical point is a point on the graph of a function where the derivative is zero or undefined, and it may represent a relative maximum, relative minimum, or point of inflection.

Concavity: Concavity refers to the curvature of a graph, and it can be used to identify relative minima and maxima as points where the graph changes from concave up to concave down, or vice versa.