Math Problem Statement

On which of the intervals does the function decrease the most?

Solution

The question asks: On which interval does the function decrease the most?

The table shows the average rate of change for different intervals:

  • 1x31 \leq x \leq 3: 1.2-1.2
  • 3x53 \leq x \leq 5: 0.2-0.2
  • 5x65 \leq x \leq 6: 2.6-2.6
  • 6x106 \leq x \leq 10: 1.6-1.6

The function decreases the most where the average rate of change is the most negative. The interval with the most negative rate of change is 5x65 \leq x \leq 6, where the rate is 2.6-2.6. This matches the answer you selected.

Let me know if you need further clarifications!

Here are 5 follow-up questions:

  1. What does the average rate of change of a function tell you about its behavior on a specific interval?
  2. How would the rate of change differ for a function that is increasing vs decreasing?
  3. How can you compute the average rate of change for any given interval if not provided?
  4. What does it mean if the rate of change is zero over an interval?
  5. How would you interpret a positive average rate of change in comparison to a negative one?

Tip: The average rate of change of a function over an interval is essentially the slope of the line connecting the endpoints of the interval on the function graph.

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Math Problem Analysis

Mathematical Concepts

Calculus
Average Rate of Change

Formulas

Average Rate of Change = (Change in Function Value) / (Change in Variable)

Theorems

Rate of Change Interpretation

Suitable Grade Level

High School - Grades 10-12