Perform a first derivative test on the function f(x) = x/9 -x;(-3,3]. a. Locate the critical points of the given function. b. Use the First Derivative Test to locate the local maximum and minimum values. c. Identify the absolute maximum and minimum values of the function on the given interval (when they exist).
Perform a first derivative test on the function f(x) = x/9 -x;(-3,3]. a. Locate the critical points of the given function. b. Use the First Derivative Test to locate the local maximum and minimum values. c. Identify the absolute maximum and minimum values of the function on the given interval (when they exist).
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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How would I solve this problem? I turn a complete blank trying to find the critical numbers, and it looks as though I need to find said numbers to complete the problem.
The first image is the problem itself, while the second image is what I had before getting stuck.
![**Perform a First Derivative Test on the Function**
Function: \( f(x) = x \sqrt{9 - x^2} \); Interval: \([-3, 3]\).
**Tasks:**
**a.** Locate the critical points of the given function.
**b.** Use the First Derivative Test to locate the local maximum and minimum values.
**c.** Identify the absolute maximum and minimum values of the function on the given interval (when they exist).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5d0110b9-0eb0-473d-8cca-8d79fb2a650a%2F57225135-4be0-436c-a162-0632356909c9%2F8a4zt98_processed.png&w=3840&q=75)
Transcribed Image Text:**Perform a First Derivative Test on the Function**
Function: \( f(x) = x \sqrt{9 - x^2} \); Interval: \([-3, 3]\).
**Tasks:**
**a.** Locate the critical points of the given function.
**b.** Use the First Derivative Test to locate the local maximum and minimum values.
**c.** Identify the absolute maximum and minimum values of the function on the given interval (when they exist).
![**Function Analysis and Derivative Calculation**
**Objective:** Analyze the function \( f(x) = x \sqrt{9 - x^2} \) and calculate its derivative.
**Domain:** \([-3, 3]\)
1. **Function Definition:**
\[ f(x) = x \sqrt{9 - x^2} \]
2. **Derivative Calculation:**
- Apply the product rule to find the derivative \( f'(x) \):
\[
f'(x) = \left( 1 \cdot \sqrt{9 - x^2} \right) + \left( x \cdot \frac{-x}{\sqrt{9 - x^2}} \right)
\]
3. **Simplified Derivative:**
\[
f'(x) = \frac{-2x^2 + 9}{\sqrt{9 - x^2}}
\]
4. **Evaluate the Derivative:**
Solving \( f'(x) = 0 \) gives:
\[
0 = \frac{-2x^2 + 9}{\sqrt{9 - x^2}}
\]
The calculations provided represent a step-by-step application of the product rule to obtain the derivative of the function, simplifying the expression for easier evaluation and analysis. The exploration of \( f'(x) = 0 \) can help identify critical points and analyze the behavior of the function on its defined interval.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5d0110b9-0eb0-473d-8cca-8d79fb2a650a%2F57225135-4be0-436c-a162-0632356909c9%2Ftad71ux_processed.png&w=3840&q=75)
Transcribed Image Text:**Function Analysis and Derivative Calculation**
**Objective:** Analyze the function \( f(x) = x \sqrt{9 - x^2} \) and calculate its derivative.
**Domain:** \([-3, 3]\)
1. **Function Definition:**
\[ f(x) = x \sqrt{9 - x^2} \]
2. **Derivative Calculation:**
- Apply the product rule to find the derivative \( f'(x) \):
\[
f'(x) = \left( 1 \cdot \sqrt{9 - x^2} \right) + \left( x \cdot \frac{-x}{\sqrt{9 - x^2}} \right)
\]
3. **Simplified Derivative:**
\[
f'(x) = \frac{-2x^2 + 9}{\sqrt{9 - x^2}}
\]
4. **Evaluate the Derivative:**
Solving \( f'(x) = 0 \) gives:
\[
0 = \frac{-2x^2 + 9}{\sqrt{9 - x^2}}
\]
The calculations provided represent a step-by-step application of the product rule to obtain the derivative of the function, simplifying the expression for easier evaluation and analysis. The exploration of \( f'(x) = 0 \) can help identify critical points and analyze the behavior of the function on its defined interval.
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