Using calculus, find the absolute maximum and absolute minimum of the function f (x) = 6x² − 36x + 2 on the interval [-5, 4]. absolute maximum = absolute minimum = M₂ HW
Using calculus, find the absolute maximum and absolute minimum of the function f (x) = 6x² − 36x + 2 on the interval [-5, 4]. absolute maximum = absolute minimum = M₂ HW
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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Question
![**Finding Absolute Extrema**
**Problem Statement:**
Using calculus, find the absolute maximum and absolute minimum of the function \( f(x) = 6x^2 - 36x + 2 \) on the interval \([-5, 4]\).
**Solution Steps:**
1. **Find Critical Points:**
- Determine the first derivative of the function \( f(x) \).
- Set the first derivative equal to zero to find the critical points.
2. **Evaluate the Function at Critical Points and Endpoints:**
- Calculate \( f(x) \) at each critical point within the given interval.
- Calculate \( f(x) \) at the endpoints of the interval \([-5, 4]\).
3. **Determination of Absolute Maximum and Minimum:**
- Compare the function values obtained from the critical points and endpoints.
- Identify the highest and lowest values to find the absolute maximum and minimum.
**Input Fields:**
- **Absolute Maximum:**
- (Input Box)
- **Absolute Minimum:**
- (Input Box)
**Graphical Explanation:**
- There are no graphs or diagrams associated with this problem.
In the input fields, students are expected to enter the values of the absolute maximum and absolute minimum identified from their calculations.
By following these steps, students can apply calculus principles appropriately to find the absolute extrema of the given quadratic function on the specified interval.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F492624c0-6418-4b29-86e0-006f11a113d1%2Fee3ac070-84c4-48aa-b934-6cf7a21326f4%2F94pmpjr_processed.png&w=3840&q=75)
Transcribed Image Text:**Finding Absolute Extrema**
**Problem Statement:**
Using calculus, find the absolute maximum and absolute minimum of the function \( f(x) = 6x^2 - 36x + 2 \) on the interval \([-5, 4]\).
**Solution Steps:**
1. **Find Critical Points:**
- Determine the first derivative of the function \( f(x) \).
- Set the first derivative equal to zero to find the critical points.
2. **Evaluate the Function at Critical Points and Endpoints:**
- Calculate \( f(x) \) at each critical point within the given interval.
- Calculate \( f(x) \) at the endpoints of the interval \([-5, 4]\).
3. **Determination of Absolute Maximum and Minimum:**
- Compare the function values obtained from the critical points and endpoints.
- Identify the highest and lowest values to find the absolute maximum and minimum.
**Input Fields:**
- **Absolute Maximum:**
- (Input Box)
- **Absolute Minimum:**
- (Input Box)
**Graphical Explanation:**
- There are no graphs or diagrams associated with this problem.
In the input fields, students are expected to enter the values of the absolute maximum and absolute minimum identified from their calculations.
By following these steps, students can apply calculus principles appropriately to find the absolute extrema of the given quadratic function on the specified interval.
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