Math Problem Statement

Which of the following statements is true about f? Based on the average rate of change of a polynomial over various intervals.

Solution

The table shows the average rates of change for a polynomial function f(x)f(x) over different intervals. Let's analyze the situation step by step to evaluate which statement is true about the graph.

The key information from the table:

  • Interval [1,2][1, 2]: Rate of change = 1.5-1.5
  • Interval [2,3][2, 3]: Rate of change = 17.517.5
  • Interval [3,4][3, 4]: Rate of change = 50.550.5
  • Interval [4,5][4, 5]: Rate of change = 98.598.5

The rate of change of a function represents its slope over a particular interval. If the rate of change is increasing, the graph of the function is concave up. If the rate of change is decreasing, the graph is concave down.

Analysis:

  • From [1,2][1, 2] to [2,3][2, 3], the rate of change increases from 1.5-1.5 to 17.517.5.
  • From [2,3][2, 3] to [3,4][3, 4], the rate of change further increases from 17.517.5 to 50.550.5.
  • From [3,4][3, 4] to [4,5][4, 5], the rate of change again increases from 50.550.5 to 98.598.5.

This consistent increase in the average rate of change suggests that the graph of ff is concave up, not concave down, over the intervals being discussed.

Evaluation of Statements:

  1. "The graph is concave up on the interval [1, 3]."

    • True. The rate of change is increasing on this interval, indicating concavity up.
  2. "The graph is concave down on the interval [1, 3]."

    • False. The graph is concave up because the rate of change is increasing between these points.
  3. "The graph is decreasing on the interval [1, 3]."

    • False. The rate of change is negative at [1,2][1, 2] but becomes positive by [2,3][2, 3], so the graph is not decreasing over the entire interval.
  4. "There is an inflection point at x=2x = 2."

    • Uncertain based on the given data. The rate of change switches from negative to positive around x=2x = 2, suggesting that there could be an inflection point, but further information would be required to confirm this.

The best choice among the options given is that the graph is concave up on the interval [1, 3], but you marked the second statement, which is incorrect.

Would you like further clarification, or do you have any other questions?


Here are 5 related questions for further exploration:

  1. What does it mean for a function to be concave up versus concave down?
  2. How can you determine whether a function has an inflection point based on rates of change?
  3. What is the relationship between the second derivative of a function and its concavity?
  4. Can a polynomial function have multiple inflection points?
  5. How does the sign of the rate of change relate to increasing and decreasing intervals?

Tip: To confirm inflection points, check where the second derivative changes sign.

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Math Problem Analysis

Mathematical Concepts

Calculus
Concavity
Polynomial Functions
Average Rate of Change

Formulas

-

Theorems

Concavity Theorem
Inflection Point Identification

Suitable Grade Level

Grades 11-12