Math Problem Statement

State the intervals where the following function is positive and negative

State the intervals where the following function is positive and negative.

Solution

The graph you've provided seems to represent a cubic or higher-degree polynomial function with turning points. To determine where the function is positive or negative, we observe its behavior relative to the x-axis.

Steps to analyze:

  1. Positive intervals: The function is positive where the graph lies above the x-axis.
  2. Negative intervals: The function is negative where the graph lies below the x-axis.

Analysis from the graph:

  • The function crosses the x-axis at approximately x=0x = 0, x=3x = 3, and x=5x = 5.
  • The function is positive between 0<x<30 < x < 3 and for x>5x > 5, as it is above the x-axis in those regions.
  • The function is negative between 3<x<53 < x < 5, where the curve lies below the x-axis.

Intervals where the function is:

  • Positive: (0,3)(5,)(0, 3) \cup (5, \infty)
  • Negative: (3,5)(3, 5)

Would you like a more detailed explanation or have any further questions?

Here are 5 related questions:

  1. What are turning points, and how do they affect the shape of polynomial graphs?
  2. How can you determine the degree of a polynomial from its graph?
  3. What does it mean when a function crosses the x-axis at multiple points?
  4. How can you use calculus to find the exact points where a function changes from positive to negative?
  5. What are the differences between intervals of increase and intervals where the function is positive?

Tip: To verify intervals analytically, finding the roots (where the function equals zero) helps break down the function's behavior.

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Math Problem Analysis

Mathematical Concepts

Polynomial Functions
Interval Analysis
Graph Interpretation

Formulas

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Theorems

Root Theorem
Sign Analysis

Suitable Grade Level

Grades 10-12