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 Statement:**
Find all the inflection points of the function.
\[ f(x) = \frac{5}{6}x^3 - 5x^2 + 9 \]
**Solution:**
To find the inflection points of the function, follow these steps:
1. **Find the second derivative \( f''(x) \):**
- Start by finding the first derivative \( f'(x) \).
- Differentiate \( f(x) = \frac{5}{6}x^3 - 5x^2 + 9 \).
- Then, find the second derivative by differentiating \( f'(x) \).
2. **Set \( f''(x) = 0 \) and solve for \( x \):**
- The values of \( x \) that satisfy this equation are potential inflection points.
3. **Determine the sign change:**
- Check intervals around each solution to see if \( f''(x) \) changes sign. This confirms the presence of an inflection point.
**Inflection Point Equation:**
\[ x = \text{(Solution Box)} \]
- Input the value(s) of \( x \) that represent inflection points in the solution box.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F718d0bec-2150-4df1-8dd3-9a5e25eabd78%2F3da02c86-d9eb-42f5-8cc0-d067913aa983%2F6se2g9j5_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find all the inflection points of the function.
\[ f(x) = \frac{5}{6}x^3 - 5x^2 + 9 \]
**Solution:**
To find the inflection points of the function, follow these steps:
1. **Find the second derivative \( f''(x) \):**
- Start by finding the first derivative \( f'(x) \).
- Differentiate \( f(x) = \frac{5}{6}x^3 - 5x^2 + 9 \).
- Then, find the second derivative by differentiating \( f'(x) \).
2. **Set \( f''(x) = 0 \) and solve for \( x \):**
- The values of \( x \) that satisfy this equation are potential inflection points.
3. **Determine the sign change:**
- Check intervals around each solution to see if \( f''(x) \) changes sign. This confirms the presence of an inflection point.
**Inflection Point Equation:**
\[ x = \text{(Solution Box)} \]
- Input the value(s) of \( x \) that represent inflection points in the solution box.
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