Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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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).](https://content.bartleby.com/qna-images/question/5d0110b9-0eb0-473d-8cca-8d79fb2a650a/57225135-4be0-436c-a162-0632356909c9/8a4zt98_processed.png)
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.](https://content.bartleby.com/qna-images/question/5d0110b9-0eb0-473d-8cca-8d79fb2a650a/57225135-4be0-436c-a162-0632356909c9/tad71ux_processed.png)
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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