Math Problem Statement

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Skip to Main content Question 1 Question 2 Question 3 Question 4 Question 5 Question 6 Question 7 Question 8 Question 9 Question 10 Question 11 Question 12 Question content area top Part 1 Use the graph of f prime and f double prime to find the critical points and inflection points of​ f, the intervals on which f is increasing and​ decreasing, and the intervals of concavity.​ Then, graph f assuming ​f(0)equals0. 4 8 12 16 x y font size decreased by 3 y equals f double prime left parenthesis x right parenthesisfont size decreased by 3 y equals f prime left parenthesis x right parenthesis

A coordinate system has a horizontal x-axis labeled from 0 to 16 in increments of 4 and a vertical y-axis. A curve labeled y = f prime (x) starts above the x-axis, falls, crossing the x-axis at 4 to a minimum at x = 8, and rises, crossing the x-axis at 12, to a maximum at 16. A curve labeled y = f double prime (x) starts at the origin, falls to a minimum at x = 4, rises, crossing the x-axis at 8, to a maximum at 12, and falls to the x-axis at 16. Question content area bottom Part 1 Find the critical points. The critical points are xequals    4 comma 12. ​(Use a comma to separate answers as​ needed.) Part 2 Find the inflection points. The inflection points occur at xequals    8. ​(Use a comma to separate answers as​ needed.) Part 3 Find the intervals on which f is increasing and decreasing. f is increasing on    left parenthesis 0 comma 4 right parenthesis comma left parenthesis 12 comma 16 right parenthesis. ​(Type your answer in interval notation. Use a comma to separate answers as​ needed.) f is decreasing on    left parenthesis 4 comma 12 right parenthesis. ​(Type your answer in interval notation. Use a comma to separate answers as​ needed.) Part 4 Find the intervals of concavity. f is concave down on    enter your response here. ​(Type your answer in interval notation. Use a comma to separate answers as​ needed.) f is concave up on    enter your response here. ​(Type your answer in interval notation. Use a comma to separate answers as​ needed.) The critical points are x(Use a comma to separate answers as needed.)The inflection points occur at x(Use a comma to separate answers as needed.)f is increasing on(Type your answer in interval notation. Use a comma to separate answers as needed.)decreasingf is decreasing on(Type your answer in interval notation. Use a comma to separate answers as needed.)f is concave down on(Type your answer in interval notation. Use a comma to separate answers as needed.)upf is concave up on(Type your answer in interval notation. Use a comma to separate answers as needed.)Choose the correct graph below.The critical points are x(Use a comma to separate answers as needed.)The inflection points occur at x(Use a comma to separate answers as needed.)f is increasing on(Type your answer in interval notation. Use a comma to separate answers as needed.)decreasingf is decreasing on(Type your answer in interval notation. Use a comma to separate answers as needed.)f is concave down on(Type your answer in interval notation. Use a comma to separate answers as needed.)upf is concave up on(Type your answer in interval notation. Use a comma to separate answers as needed.) Textbook Calculator

Solution

To determine the intervals of concavity, let’s examine the behavior of f′′(x)f''(x):

  1. Concavity Determination:

    • Since f(x)f(x) is concave up where f′′(x)>0f''(x) > 0 and concave down where f′′(x)<0f''(x) < 0, let's use the zero crossings and the changes in sign of f′′(x)f''(x) at x=4,8,x = 4, 8, and x=12x = 12.

    • From the problem, we have:

      • f′′(x)<0f''(x) < 0 on (0,8)(0, 8): Concave down on (0,8)(0, 8)
      • f′′(x)>0f''(x) > 0 on (8,16)(8, 16): Concave up on (8,16)(8, 16)

  2. Final Answers:

    • Concave down on (0,8)(0, 8)
    • Concave up on (8,16)(8, 16)

Let me know if you would like further clarification on any part of this, or if you have questions about graphing f(x)f(x).

Here are five follow-up questions to deepen your understanding:

  1. Why is concavity important in determining the shape of f(x)f(x)?
  2. How do we identify concavity from the sign of f′′(x)f''(x)?
  3. What effect do critical points have on the intervals where f(x)f(x) is increasing or decreasing?
  4. How would the intervals change if f′′(x)f''(x) had additional zero crossings?
  5. How could you use f′(x)f'(x) and f′′(x)f''(x) graphs together to sketch f(x)f(x)?

Tip: Always examine both the sign changes and zero crossings of f′′(x)f''(x) to accurately determine intervals of concavity.

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Math Problem Analysis

Mathematical Concepts

Calculus
Critical Points
Inflection Points
Intervals of Increase and Decrease
Concavity

Formulas

f' = 0 at critical points
f'' > 0 indicates concave up
f'' < 0 indicates concave down

Theorems

First Derivative Test
Second Derivative Test

Suitable Grade Level

Grades 11-12

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