Math Problem Statement

The function graphed above is:

Increasing on the interval(s) _________

Decreasing on the interval(s) ___________

The function graphed above is: Increasing on the interval(s) _________ Decreasing on the interval(s) ___________

Solution

From the graph, we can observe the behavior of the function based on where the slope is positive (increasing) and where it is negative (decreasing).

Increasing Intervals:

The function is increasing where the slope of the graph is positive (upward). This occurs between:

  1. (,2)(-\infty, -2)
  2. (0,2)(0, 2)

Decreasing Intervals:

The function is decreasing where the slope of the graph is negative (downward). This occurs between:

  1. (2,0)(-2, 0)
  2. (2,)(2, \infty)

To summarize:

  • Increasing on: (,2)(0,2)(-\infty, -2) \cup (0, 2)
  • Decreasing on: (2,0)(2,)(-2, 0) \cup (2, \infty)

Would you like more details or explanations on this topic?

Here are 5 related questions:

  1. How do you determine critical points of a function?
  2. What are the steps to find increasing and decreasing intervals algebraically?
  3. How do you calculate the first derivative to analyze function behavior?
  4. What is the significance of local maxima and minima in a graph?
  5. How can you classify a function's concavity from its graph?

Tip: A function's first derivative helps you determine where it's increasing or decreasing.

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Math Problem Analysis

Mathematical Concepts

Calculus
Function Behavior
Graph Analysis

Formulas

Derivative to find critical points
Analysis of intervals using the first derivative

Theorems

First Derivative Test

Suitable Grade Level

Grades 11-12