Math Problem Statement

Consider the quartic polynomial function f graphed here. Where is it decreasing? (Write your answer using interval notation.)

Solution

To determine where the quartic polynomial function is decreasing from the graph, we need to identify the intervals where the slope of the graph is negative (i.e., the graph is moving downwards).

Observing the Graph:

  1. The function appears to decrease from approximately:
    • Left side: The graph decreases from around x=2.5x = -2.5 to x=1x = -1.
    • Right side: The graph also decreases from around x=1.5x = 1.5 to x=3x = 3.

Writing in Interval Notation:

  • The function is decreasing on two intervals: (2.5,1)(1.5,3)(-2.5, -1) \cup (1.5, 3)

This is the answer in interval notation.

Would you like further details or have any questions?

Here are 5 related questions to explore further:

  1. What is the difference between increasing and decreasing intervals of a function?
  2. How can you determine where a function is increasing or decreasing by using the derivative?
  3. What are critical points, and how do they relate to increasing and decreasing intervals?
  4. How does the behavior of a polynomial function change at its local maxima and minima?
  5. Can a function be both increasing and decreasing at the same point?

Tip: When reading a graph, always look for intervals where the curve slopes downwards to identify where the function is decreasing.

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Math Problem Analysis

Mathematical Concepts

Calculus
Polynomial Functions
Interval Notation

Formulas

Identifying intervals where a function is increasing or decreasing by examining the slope of the function

Theorems

Fundamental Theorem of Algebra

Suitable Grade Level

Grades 11-12