Math Problem Statement

Solve it

Consider the graph of f. Label whether the graph is increasing or decreasing and whether the graph is concave up or down on each interval. Then, determine what these characteristics imply about the first and second derivative of f on each interval.

Solution

To solve this problem, we need to analyze the behavior of the function ff across each interval AA, BB, CC, and DD. We will determine whether the function is increasing or decreasing, if it is concave up or down, and the signs of the first and second derivatives, ff' and ff'', respectively, based on these observations.

Analysis of Each Interval

  1. Interval A (from the start of the graph to point B):

    • The graph is increasing (since it moves upwards as xx increases).
    • The curve is concave down (it forms a downward "U" shape).
    • Sign of ff': Positive (because the graph is increasing).
    • Sign of ff'': Negative (because the graph is concave down).
  2. Interval B (between points B and C):

    • The graph is decreasing (it moves downward as xx increases).
    • The curve is concave down.
    • Sign of ff': Negative (because the graph is decreasing).
    • Sign of ff'': Negative (since it remains concave down).
  3. Interval C (between points C and D):

    • The graph is decreasing.
    • The curve is concave up (forms an upward "U" shape).
    • Sign of ff': Negative (since it’s still decreasing).
    • Sign of ff'': Positive (as it is concave up).
  4. Interval D (after point D):

    • The graph is increasing.
    • The curve is concave up.
    • Sign of ff': Positive (because the graph is increasing).
    • Sign of ff'': Positive (since it is concave up).

Solution Summary Table

Interval AInterval BInterval CInterval D
Increasing or DecreasingIncreasingDecreasingDecreasingIncreasing
Concave Up or DownConcave DownConcave DownConcave UpConcave Up
Sign of ff'PositiveNegativeNegativePositive
Sign of ff''NegativeNegativePositivePositive

Would you like more detailed explanations on each interval or have any questions?

Related Questions

  1. How do concavity and the second derivative relate in general?
  2. What does it mean when the first derivative ff' changes signs?
  3. Why does concavity affect the sign of the second derivative?
  4. How can you determine if a critical point is a maximum or minimum?
  5. What does the graph tell us about inflection points and their relationship to f=0f'' = 0?

Tip: Remember, the first derivative ff' tells us if the function is increasing or decreasing, while the second derivative ff'' indicates the concavity.

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Math Problem Analysis

Mathematical Concepts

Calculus
Graph Analysis
Derivatives
Concavity

Formulas

First derivative f' indicates increasing or decreasing behavior
Second derivative f'' indicates concavity

Theorems

The First Derivative Test
The Second Derivative Test

Suitable Grade Level

Grades 11-12