Math Problem Statement

a) find the open intervals on which the function is increasing and those on which it is decreasing.

find the open intervals on which the function is increasing and those on which it is decreasing.

Solution

Let's begin by analyzing the given graph of the function to determine where it is increasing and where it is decreasing. Based on the graph, we can identify key regions where the slope of the function is positive (increasing) or negative (decreasing).

Approach:

  1. Increasing Interval: A function is increasing where its graph moves upwards as you go from left to right (positive slope).
  2. Decreasing Interval: A function is decreasing where its graph moves downwards as you go from left to right (negative slope).

Let me now analyze the graph and provide the intervals for increasing and decreasing behavior.Upon reviewing the graph, I can identify the following behavior of the function:

1. Increasing Intervals:

The function is increasing where the graph has a positive slope (rising as we move left to right). Based on the visual inspection of the graph:

  • The function increases on the interval (,2)(-\infty, -2).
  • The function also increases on the interval (1,3)(1, 3).

2. Decreasing Intervals:

The function is decreasing where the graph has a negative slope (falling as we move left to right):

  • The function decreases on the interval (2,1)(-2, 1).
  • The function decreases on the interval (3,)(3, \infty).

Summary of Intervals:

  • Increasing: (,2)(-\infty, -2) and (1,3)(1, 3)
  • Decreasing: (2,1)(-2, 1) and (3,)(3, \infty)

Would you like a more detailed explanation of how these intervals were identified, or do you have any other questions?

Related questions:

  1. How do the first and second derivatives help in determining increasing and decreasing intervals?
  2. What are the critical points, and how are they determined from the graph?
  3. How does concavity affect the behavior of a function in these intervals?
  4. Can we verify the intervals using calculus techniques (derivatives)?
  5. How do we identify points of inflection from the graph?

Tip: Always check for turning points or critical points where the graph changes direction, as they often mark where the function changes from increasing to decreasing, or vice versa.

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Math Problem Analysis

Mathematical Concepts

Calculus
Function Analysis
Intervals
Increasing and Decreasing Functions

Formulas

None (relies on graph analysis)

Theorems

First derivative test
Concept of slope of a function

Suitable Grade Level

Grades 11-12