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use derivatives to determine where f(x) = x³ - 2x² + x +1 (a) where function is increasing and decreasing (b) has local minima and maxima (if any) (c) is concave up and concave down; (a) has inflection points.

use derivatives to determine where f(x) = x³ - 2x² + x +1 (a) where function is increasing and decreasing (b) has local minima and maxima (if any) (c) is concave up and concave down; (a) has inflection points.

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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**Title: Understanding Function Behavior Using Derivatives** **Objective:** Analyze the function \( f(x) = x^3 - 2x^2 + x + 1 \) using derivatives to determine: 1. Where the function is increasing and decreasing. 2. Points of local minima and maxima (if any). 3. Intervals where the function is concave up and concave down. 4. Points of inflection. **Instructions:** - Show your work for each part of the analysis. **Steps to Analyze the Function:** (a) **Determine where the function is increasing and decreasing.** - Find the first derivative of the function \( f(x) \). - Set the first derivative equal to zero and solve for \( x \) to find critical points. - Use the first derivative test to determine intervals of increase and decrease. (b) **Identify local minima and maxima.** - Use the critical points obtained from part (a). - Apply the first or second derivative test to classify the critical points as local minima, maxima, or neither. (c) **Determine concave up and concave down intervals.** - Find the second derivative of the function \( f(x) \). - Set the second derivative equal to zero and solve for \( x \) to find potential inflection points. - Use the second derivative test to determine intervals of concavity. (d) **Identify inflection points.** - Verify the points where the second derivative changes sign. - These points, where concavity changes, are the inflection points.
Transcribed Image Text:**Title: Understanding Function Behavior Using Derivatives** **Objective:** Analyze the function \( f(x) = x^3 - 2x^2 + x + 1 \) using derivatives to determine: 1. Where the function is increasing and decreasing. 2. Points of local minima and maxima (if any). 3. Intervals where the function is concave up and concave down. 4. Points of inflection. **Instructions:** - Show your work for each part of the analysis. **Steps to Analyze the Function:** (a) **Determine where the function is increasing and decreasing.** - Find the first derivative of the function \( f(x) \). - Set the first derivative equal to zero and solve for \( x \) to find critical points. - Use the first derivative test to determine intervals of increase and decrease. (b) **Identify local minima and maxima.** - Use the critical points obtained from part (a). - Apply the first or second derivative test to classify the critical points as local minima, maxima, or neither. (c) **Determine concave up and concave down intervals.** - Find the second derivative of the function \( f(x) \). - Set the second derivative equal to zero and solve for \( x \) to find potential inflection points. - Use the second derivative test to determine intervals of concavity. (d) **Identify inflection points.** - Verify the points where the second derivative changes sign. - These points, where concavity changes, are the inflection points.
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