Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function. 30 X y = 3x² +
Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function. 30 X y = 3x² +
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Problem Statement:**
Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function.
\[ y = 3x^2 + \frac{30}{x} \]
**Solution Guide:**
1. **Finding Critical Points:**
- Calculate the derivative of the function \( y = 3x^2 + \frac{30}{x} \) to find critical points.
- Set the derivative to zero and solve for \( x \).
2. **Determining Local and Absolute Extrema:**
- Use the second derivative test to classify the nature (local maxima/minima) of the critical points.
- Evaluate the function at any critical points and endpoints to find absolute extrema.
3. **Finding Inflection Points:**
- Calculate the second derivative of the function.
- Set the second derivative to zero and solve for \( x \) to find potential inflection points.
- Confirm the change in concavity to identify inflection points.
4. **Graphing the Function:**
- Plot the function, marking critical points, inflection points, and extrema.
- Use a graphing tool to visualize and confirm your findings.
**Graph Explanation:**
- The graph should display the curve of the function \( y = 3x^2 + \frac{30}{x} \).
- Critical points should be indicated, showing where the derivative is zero.
- Mark inflection points to demonstrate where the curve changes concavity.
- Highlight areas of local and absolute maxima and minima for clarity.
This process will help you understand the behavior of the function and identify key characteristics such as growth, decay, and curvature changes.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F60e5fca3-b9c2-47c4-9eba-5478d8641133%2F52ddb4b6-dfa8-4b33-8263-aa1335b57125%2Fw8vn8ry_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function.
\[ y = 3x^2 + \frac{30}{x} \]
**Solution Guide:**
1. **Finding Critical Points:**
- Calculate the derivative of the function \( y = 3x^2 + \frac{30}{x} \) to find critical points.
- Set the derivative to zero and solve for \( x \).
2. **Determining Local and Absolute Extrema:**
- Use the second derivative test to classify the nature (local maxima/minima) of the critical points.
- Evaluate the function at any critical points and endpoints to find absolute extrema.
3. **Finding Inflection Points:**
- Calculate the second derivative of the function.
- Set the second derivative to zero and solve for \( x \) to find potential inflection points.
- Confirm the change in concavity to identify inflection points.
4. **Graphing the Function:**
- Plot the function, marking critical points, inflection points, and extrema.
- Use a graphing tool to visualize and confirm your findings.
**Graph Explanation:**
- The graph should display the curve of the function \( y = 3x^2 + \frac{30}{x} \).
- Critical points should be indicated, showing where the derivative is zero.
- Mark inflection points to demonstrate where the curve changes concavity.
- Highlight areas of local and absolute maxima and minima for clarity.
This process will help you understand the behavior of the function and identify key characteristics such as growth, decay, and curvature changes.
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