6. Let y = 2x³ - ²x²¹ - 10x³ + 2019 Find the interval(s) where the function is concave up and interval(s) where the function is concave down. Identify the x-value of any inflection point(s).
6. Let y = 2x³ - ²x²¹ - 10x³ + 2019 Find the interval(s) where the function is concave up and interval(s) where the function is concave down. Identify the x-value of any inflection point(s).
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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![### Educational Content on Calculus: Analyzing Functions
---
#### 6. Polynomial Function Analysis
Let
\[ y = 2x^5 - \frac{5}{3}x^4 - 10x^3 + 2019 \]
**Task:**
- Find the interval(s) where the function is concave up.
- Find the interval(s) where the function is concave down.
- Identify the \( x \)-value(s) of any inflection point(s).
---
#### 7. Analysis of a Continuous Function
Consider a function \( f(x) \) which is defined and continuous on the interval \( 0 \leq x \leq 8 \). Answer the following questions for the function \( f(x) \). The graph of \( f'(x) \), the derivative of \( f \), is provided for context.
**a)** Given the graph of \( f'(x) \) (the derivative of \( f(x) \)), sketch \( f''(x) \) on the provided axis on the right. Any curve with the appropriate zeroes, signs, and discontinuities of \( f''(x) \) will suffice.
**Graph Explanation:**
- The graph of \( y = f'(x) \) shows a curve with:
- Zeroes at \( x = 2 \) and \( x = 6 \).
- A local minimum just before \( x = 4 \) and a local maximum around \( x = 7 \).
- A discontinuity as \( x \) approaches 8 from the left.
**b)** Identify the interval(s) where \( f(x) \) is increasing and where \( f(x) \) is decreasing.
**c)** Identify the \( x \)-coordinates for critical points of \( f(x) \). Classify each critical point.
**d)** Identify the interval(s) where \( f(x) \) is concave up and where \( f(x) \) is concave down.
**e)** Identify the \( x \)-coordinates for all inflection points of \( f(x) \).
---
This exercise helps students practice identifying key characteristics of functions through their derivatives, enhancing their understanding of calculus concepts such as concavity, critical points, and inflection points.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa82ab235-0403-49ff-b89c-b564bdf9f108%2Fbed7b9ea-b3f8-4f05-981d-75098c247804%2Foqh6gci_processed.png&w=3840&q=75)
Transcribed Image Text:### Educational Content on Calculus: Analyzing Functions
---
#### 6. Polynomial Function Analysis
Let
\[ y = 2x^5 - \frac{5}{3}x^4 - 10x^3 + 2019 \]
**Task:**
- Find the interval(s) where the function is concave up.
- Find the interval(s) where the function is concave down.
- Identify the \( x \)-value(s) of any inflection point(s).
---
#### 7. Analysis of a Continuous Function
Consider a function \( f(x) \) which is defined and continuous on the interval \( 0 \leq x \leq 8 \). Answer the following questions for the function \( f(x) \). The graph of \( f'(x) \), the derivative of \( f \), is provided for context.
**a)** Given the graph of \( f'(x) \) (the derivative of \( f(x) \)), sketch \( f''(x) \) on the provided axis on the right. Any curve with the appropriate zeroes, signs, and discontinuities of \( f''(x) \) will suffice.
**Graph Explanation:**
- The graph of \( y = f'(x) \) shows a curve with:
- Zeroes at \( x = 2 \) and \( x = 6 \).
- A local minimum just before \( x = 4 \) and a local maximum around \( x = 7 \).
- A discontinuity as \( x \) approaches 8 from the left.
**b)** Identify the interval(s) where \( f(x) \) is increasing and where \( f(x) \) is decreasing.
**c)** Identify the \( x \)-coordinates for critical points of \( f(x) \). Classify each critical point.
**d)** Identify the interval(s) where \( f(x) \) is concave up and where \( f(x) \) is concave down.
**e)** Identify the \( x \)-coordinates for all inflection points of \( f(x) \).
---
This exercise helps students practice identifying key characteristics of functions through their derivatives, enhancing their understanding of calculus concepts such as concavity, critical points, and inflection points.
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