Math Problem Statement

Use the graph to determine the open intervals over which f left parenthesis x right parenthesis is increasing, decreasing, or constant, then determine all the local minimum and maximum values on the graph.

A.) Decreasing on left parenthesis short dash infinity comma space 0 right parenthesis union left parenthesis 1 comma space 2 right parenthesis Increasing on left parenthesis 0 comma space 1 right parenthesis union left parenthesis 2 comma space infinity right parenthesis Local minimum is 0 at x equals 0 and 2 at x equals 2 Local maximum is 1 at x equals 1

B.) Decreasing on left parenthesis short dash infinity comma space 0 right square bracket union left square bracket 1 comma space 2 right parenthesis Increasing on left square bracket 0 comma space 1 right square bracket union left square bracket 2 comma space infinity right parenthesis Local minimum is -2 at x equals 0 and x equals 2 Local maximum is -1 at x equals 1

C.) Decreasing on left parenthesis short dash infinity comma space 0 right square bracket union left square bracket 1 comma space 2 right square bracket Increasing on left square bracket 0 comma space 1 right square bracket union left square bracket 2 comma space infinity right parenthesis Local minimum is 0 at x equals 0 and 2 at x equals 2 Local maximum is 1 at x equals 1

D.) Decreasing on left parenthesis short dash infinity comma space 0 right parenthesis union left parenthesis 1 comma space 2 right parenthesis Increasing on left parenthesis 0 comma space 1 right parenthesis union left parenthesis 2 comma infinity right parenthesis Local minimum is -2 at x equals 0 and x equals 2 Local maximum is -1 at x equals 1

Use the graph to determine the open intervals over which f(x) is increasing, decreasing, or constant, then determine all the local minimum and maximum values on the graph.

Solution

To analyze the graph and determine the correct intervals where f(x)f(x) is increasing, decreasing, or constant, along with identifying the local minima and maxima, I will interpret the graph in the uploaded image.

Observations:

  • The graph appears to decrease as it approaches x=0x = 0, reaching a local minimum at (0,−2)(0, -2).
  • It then increases from x=0x = 0 to x=1x = 1, reaching a local maximum at (1,−1)(1, -1).
  • From x=1x = 1 to x=2x = 2, the graph decreases again, reaching another local minimum at (2,−2)(2, -2).
  • Finally, from x=2x = 2 onward, the graph increases.

Analysis of Intervals:

  • Decreasing: (−∞,0]∪[1,2](-\infty, 0] \cup [1, 2]
  • Increasing: [0,1]∪[2,∞)[0, 1] \cup [2, \infty)

Local Minima:

  • At x=0x = 0, the minimum value is −2-2.
  • At x=2x = 2, the minimum value is −2-2.

Local Maximum:

  • At x=1x = 1, the maximum value is −1-1.

Correct Answer:

The correct option is:

B.)

  • Decreasing on (−∞,0]∪[1,2)(-\infty, 0] \cup [1, 2)
  • Increasing on [0,1]∪[2,∞)[0, 1] \cup [2, \infty)
  • Local minimum is −2-2 at x=0x = 0 and x=2x = 2.
  • Local maximum is −1-1 at x=1x = 1.

Would you like further clarification or a detailed step-by-step explanation?

Here are 5 questions to deepen your understanding:

  1. What does it mean for a function to be "increasing" or "decreasing" on an interval?
  2. How do you determine local minima and maxima from a graph?
  3. Can a function have more than one local minimum or maximum?
  4. What are the critical points of the given function?
  5. How can you use calculus to confirm these intervals of increase or decrease?

Tip: Always check the endpoints of intervals carefully when determining where the function is increasing or decreasing!

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Math Problem Analysis

Mathematical Concepts

Graph Analysis
Intervals of Increase and Decrease
Local Extrema

Formulas

No specific formula, but graph-based interval analysis

Theorems

First Derivative Test (conceptual) for increasing and decreasing intervals

Suitable Grade Level

Grades 9-12

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