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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![**Problem 3: Local Minima and Maxima Using the First Derivative Test**
Objective: Use the first derivative test to determine all local minima and maxima for the following function:
\[ f(x) = 4x^3 - 9x^2 - 30x + 6 \]
Instructions:
1. **Find the first derivative** of the function \( f(x) \).
2. **Identify critical points** by setting the first derivative equal to zero and solving for \( x \).
3. **Use the first derivative test** to classify each critical point as a local minimum, local maximum, or neither.
Explanation:
- The first derivative, \( f'(x) \), provides information about the slope of the tangent line to the curve at any point \( x \).
- Critical points occur where \( f'(x) = 0 \) or where \( f'(x) \) is undefined.
- The first derivative test involves checking the sign of \( f'(x) \) on intervals around each critical point to determine if the function is increasing or decreasing. This helps classify the nature of each critical point.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff7da7361-3083-4e81-80b4-424dbe239c3a%2F81e0ac34-2b0b-4d39-835c-9d77c95b3e96%2F1dmly8p_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem 3: Local Minima and Maxima Using the First Derivative Test**
Objective: Use the first derivative test to determine all local minima and maxima for the following function:
\[ f(x) = 4x^3 - 9x^2 - 30x + 6 \]
Instructions:
1. **Find the first derivative** of the function \( f(x) \).
2. **Identify critical points** by setting the first derivative equal to zero and solving for \( x \).
3. **Use the first derivative test** to classify each critical point as a local minimum, local maximum, or neither.
Explanation:
- The first derivative, \( f'(x) \), provides information about the slope of the tangent line to the curve at any point \( x \).
- Critical points occur where \( f'(x) = 0 \) or where \( f'(x) \) is undefined.
- The first derivative test involves checking the sign of \( f'(x) \) on intervals around each critical point to determine if the function is increasing or decreasing. This helps classify the nature of each critical point.
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