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Find the largest open interval where the function is changing as requested. 24) Increasing f(x)=x2-2x+1

Find the largest open interval where the function is changing as requested. 24) Increasing f(x)=x2-2x+1

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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### Finding Intervals of Increasing and Decreasing Functions In this section, we will determine the largest open intervals where given functions are either increasing or decreasing. #### Problem 24: Increasing **Function:** \( f(x) = x^2 - 2x + 1 \) To find where the function is increasing, we will: 1. Calculate the first derivative \( f'(x) \). 2. Determine where \( f'(x) > 0 \). #### Problem 25: Increasing **Function:** \( f(x) = \frac{1}{x^2 + 1} \) Steps to find where the function is increasing: 1. Calculate the first derivative \( f'(x) \). 2. Identify the interval where \( f'(x) > 0 \). #### Problem 26: Decreasing **Function:** \( f(x) = x^3 - 4x \) To find where the function is decreasing, follow these steps: 1. Calculate the first derivative \( f'(x) \). 2. Determine the interval where \( f'(x) < 0 \). #### Problem 27: Decreasing **Function:** \( f(x) = \sqrt{4 - x} \) To determine the interval where this function is decreasing: 1. Find the first derivative \( f'(x) \). 2. Identify where \( f'(x) < 0 \). By following these steps for each function, you will be able to pinpoint the largest open intervals where each function is changing as specified.
Transcribed Image Text:### Finding Intervals of Increasing and Decreasing Functions In this section, we will determine the largest open intervals where given functions are either increasing or decreasing. #### Problem 24: Increasing **Function:** \( f(x) = x^2 - 2x + 1 \) To find where the function is increasing, we will: 1. Calculate the first derivative \( f'(x) \). 2. Determine where \( f'(x) > 0 \). #### Problem 25: Increasing **Function:** \( f(x) = \frac{1}{x^2 + 1} \) Steps to find where the function is increasing: 1. Calculate the first derivative \( f'(x) \). 2. Identify the interval where \( f'(x) > 0 \). #### Problem 26: Decreasing **Function:** \( f(x) = x^3 - 4x \) To find where the function is decreasing, follow these steps: 1. Calculate the first derivative \( f'(x) \). 2. Determine the interval where \( f'(x) < 0 \). #### Problem 27: Decreasing **Function:** \( f(x) = \sqrt{4 - x} \) To determine the interval where this function is decreasing: 1. Find the first derivative \( f'(x) \). 2. Identify where \( f'(x) < 0 \). By following these steps for each function, you will be able to pinpoint the largest open intervals where each function is changing as specified.
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