### Sketch a Graph of \( f(x) \) Satisfying the Given Conditions Given: - \( f(0) = 2 \) and \( f(-1) = 0 \) - \(\lim_{{x \to -\infty}} f(x) = -3\) - \(\lim_{{x \to 3^-}} f(x) = -\infty\) Derivative Conditions: - \( f'(x) > 0 \) when \( x \in (-\infty, 0) \) - \( f'(x) < 0 \) when \( x \in (0, 3) \) and \( x \in (3, \infty) \) Second Derivative Conditions: - \( f''(x) > 0 \) when \( x \in (-\infty, -1) \) and \( x \in (3, \infty) \) - \( f''(x) < 0 \) when \( x \in (-1, 3) \) #### Graph and Explanation: The conditions provided describe how the function \( f(x) \) behaves at certain points and intervals. Below is how you can interpret and visualize these conditions on a graph: 1. **Points and Limits:** - \( f(0) = 2 \): The function passes through the point (0, 2). - \( f(-1) = 0 \): The function passes through the point (-1, 0). - \(\lim_{{x \to -\infty}} f(x) = -3\): As \( x \) approaches \( -\infty \), \( f(x) \) approaches -3. - \(\lim_{{x \to 3^-}} f(x) = -\infty\): As \( x \) approaches 3 from the left, \( f(x) \) goes to \( -\infty \). 2. **First Derivative (\( f'(x) \)):** - \( f'(x) > 0 \) when \( x \in (-\infty, 0) \): The function is increasing for \( x \) less than 0. - \( f'(x) < 0 \) when \( x \in (0, 3) \) and \(
### Sketch a Graph of \( f(x) \) Satisfying the Given Conditions Given: - \( f(0) = 2 \) and \( f(-1) = 0 \) - \(\lim_{{x \to -\infty}} f(x) = -3\) - \(\lim_{{x \to 3^-}} f(x) = -\infty\) Derivative Conditions: - \( f'(x) > 0 \) when \( x \in (-\infty, 0) \) - \( f'(x) < 0 \) when \( x \in (0, 3) \) and \( x \in (3, \infty) \) Second Derivative Conditions: - \( f''(x) > 0 \) when \( x \in (-\infty, -1) \) and \( x \in (3, \infty) \) - \( f''(x) < 0 \) when \( x \in (-1, 3) \) #### Graph and Explanation: The conditions provided describe how the function \( f(x) \) behaves at certain points and intervals. Below is how you can interpret and visualize these conditions on a graph: 1. **Points and Limits:** - \( f(0) = 2 \): The function passes through the point (0, 2). - \( f(-1) = 0 \): The function passes through the point (-1, 0). - \(\lim_{{x \to -\infty}} f(x) = -3\): As \( x \) approaches \( -\infty \), \( f(x) \) approaches -3. - \(\lim_{{x \to 3^-}} f(x) = -\infty\): As \( x \) approaches 3 from the left, \( f(x) \) goes to \( -\infty \). 2. **First Derivative (\( f'(x) \)):** - \( f'(x) > 0 \) when \( x \in (-\infty, 0) \): The function is increasing for \( x \) less than 0. - \( f'(x) < 0 \) when \( x \in (0, 3) \) and \(
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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Transcribed Image Text:### Sketch a Graph of \( f(x) \) Satisfying the Given Conditions
Given:
- \( f(0) = 2 \) and \( f(-1) = 0 \)
- \(\lim_{{x \to -\infty}} f(x) = -3\)
- \(\lim_{{x \to 3^-}} f(x) = -\infty\)
Derivative Conditions:
- \( f'(x) > 0 \) when \( x \in (-\infty, 0) \)
- \( f'(x) < 0 \) when \( x \in (0, 3) \) and \( x \in (3, \infty) \)
Second Derivative Conditions:
- \( f''(x) > 0 \) when \( x \in (-\infty, -1) \) and \( x \in (3, \infty) \)
- \( f''(x) < 0 \) when \( x \in (-1, 3) \)
#### Graph and Explanation:
The conditions provided describe how the function \( f(x) \) behaves at certain points and intervals. Below is how you can interpret and visualize these conditions on a graph:
1. **Points and Limits:**
- \( f(0) = 2 \): The function passes through the point (0, 2).
- \( f(-1) = 0 \): The function passes through the point (-1, 0).
- \(\lim_{{x \to -\infty}} f(x) = -3\): As \( x \) approaches \( -\infty \), \( f(x) \) approaches -3.
- \(\lim_{{x \to 3^-}} f(x) = -\infty\): As \( x \) approaches 3 from the left, \( f(x) \) goes to \( -\infty \).
2. **First Derivative (\( f'(x) \)):**
- \( f'(x) > 0 \) when \( x \in (-\infty, 0) \): The function is increasing for \( x \) less than 0.
- \( f'(x) < 0 \) when \( x \in (0, 3) \) and \(
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