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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![## Calculus Exercise: Finding Critical Points and Absolute Extrema
### Problem Statement
For the function \( f(x) = 2x^4 - 4x^2 \) defined on the interval \(-2 \le x \le 3\):
A. **Find all critical points.**
B. **Find the absolute maximum and minimum.**
### Instructions
To solve this problem, follow these steps:
1. **Find the first derivative \( f'(x) \) of the function.**
2. **Set the first derivative equal to zero and solve for \( x \) to find the critical points.**
3. **Evaluate \( f(x) \) at the critical points and at the endpoints of the interval to determine the absolute maximum and minimum.**
### Solutions Outline
**Step 1: Calculate the first derivative \( f'(x) \).**
Given:
\[ f(x) = 2x^4 - 4x^2 \]
**Step 2: Find the critical points by setting \( f'(x) = 0 \).**
**Step 3: Evaluate \( f(x) \) at critical points and at the endpoints \( x = -2 \) and \( x = 3 \).**
Make sure to practice showing all your work and simplifications clearly, as this is crucial for understanding and verifying your answers.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4ba035d3-48b5-40e3-a0ff-fae4c289324e%2F62a7a3cd-3d90-4cf3-872f-96a9a4894007%2Fi5h0m8_processed.jpeg&w=3840&q=75)
Transcribed Image Text:## Calculus Exercise: Finding Critical Points and Absolute Extrema
### Problem Statement
For the function \( f(x) = 2x^4 - 4x^2 \) defined on the interval \(-2 \le x \le 3\):
A. **Find all critical points.**
B. **Find the absolute maximum and minimum.**
### Instructions
To solve this problem, follow these steps:
1. **Find the first derivative \( f'(x) \) of the function.**
2. **Set the first derivative equal to zero and solve for \( x \) to find the critical points.**
3. **Evaluate \( f(x) \) at the critical points and at the endpoints of the interval to determine the absolute maximum and minimum.**
### Solutions Outline
**Step 1: Calculate the first derivative \( f'(x) \).**
Given:
\[ f(x) = 2x^4 - 4x^2 \]
**Step 2: Find the critical points by setting \( f'(x) = 0 \).**
**Step 3: Evaluate \( f(x) \) at critical points and at the endpoints \( x = -2 \) and \( x = 3 \).**
Make sure to practice showing all your work and simplifications clearly, as this is crucial for understanding and verifying your answers.
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