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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![### Polynomial Root Identification
Given the polynomial function:
\[ f(x) = 10x^2 - 37x + 11 \]
We are asked to find two real numbers \( a \) and \( b \) such that \( a < b \) and:
\[ f(a) = 4 \quad \text{and} \quad f(b) = 4 \]
Additionally, we must find a real number \( c \) within the interval \((a, b)\) such that:
\[ f(c) = 0 \]
### Explanation
To solve the given problem:
1. **Identify \( a \) and \( b \):** Determine the two points where the polynomial equals 4.
2. **Find \( c \):** Use the Intermediate Value Theorem (IVT), which states that if \( f \) is continuous on the interval \([a, b]\) and \( f(a) \neq f(b) \), then there is at least one \( c \) in \((a, b)\) such that \( f(c) = 0 \).
### Equations:
- Function: \( f(x) = 10x^2 - 37x + 11 \)
- Points to find: \( f(a) = 4 \), \( f(b) = 4 \)
- Interval for \( c \): \( (a, b) \)
- Root within interval: \( f(c) = 0 \)
Ensure to solve these equations accurately to find the requested values.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F57187003-fb16-4d74-860f-bf23fec3e171%2F56fbf580-c045-4678-b902-0fc87455b978%2Fwyxuy17_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Polynomial Root Identification
Given the polynomial function:
\[ f(x) = 10x^2 - 37x + 11 \]
We are asked to find two real numbers \( a \) and \( b \) such that \( a < b \) and:
\[ f(a) = 4 \quad \text{and} \quad f(b) = 4 \]
Additionally, we must find a real number \( c \) within the interval \((a, b)\) such that:
\[ f(c) = 0 \]
### Explanation
To solve the given problem:
1. **Identify \( a \) and \( b \):** Determine the two points where the polynomial equals 4.
2. **Find \( c \):** Use the Intermediate Value Theorem (IVT), which states that if \( f \) is continuous on the interval \([a, b]\) and \( f(a) \neq f(b) \), then there is at least one \( c \) in \((a, b)\) such that \( f(c) = 0 \).
### Equations:
- Function: \( f(x) = 10x^2 - 37x + 11 \)
- Points to find: \( f(a) = 4 \), \( f(b) = 4 \)
- Interval for \( c \): \( (a, b) \)
- Root within interval: \( f(c) = 0 \)
Ensure to solve these equations accurately to find the requested values.
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