Sketch the graph of a single function f(x) that satisfies all of the following conditions, labeling all local extrema, inflection points, and asymptotes. Afterwards, explicitly state the intervals of increase, decrease, and concavity. • The domain of f(x) is (-∞, ∞) ƒ'(x) = 5x²(x² − 3) f"(x) = 10x (2x² – 3) ● ● f(0) = 0
Sketch the graph of a single function f(x) that satisfies all of the following conditions, labeling all local extrema, inflection points, and asymptotes. Afterwards, explicitly state the intervals of increase, decrease, and concavity. • The domain of f(x) is (-∞, ∞) ƒ'(x) = 5x²(x² − 3) f"(x) = 10x (2x² – 3) ● ● f(0) = 0
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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![### Educational Resource: Graphing Functions with Specific Conditions
#### Task Description:
Sketch the graph of a single function \( f(x) \) that satisfies all of the following conditions, labeling all local extrema, inflection points, and asymptotes.
#### Steps to Follow:
1. **Identify Key Features**:
- **Local Extrema**: Points where the function changes from increasing to decreasing or vice versa.
- **Inflection Points**: Points where the concavity of the function changes.
- **Asymptotes**: Lines that the graph approaches but does not touch.
2. **Explicitly State the Intervals**:
- Intervals of increase and decrease.
- Intervals of concavity.
### Given Conditions:
1. **The domain of \( f(x) \)** is \( (-\infty, \infty) \).
2. The first derivative of the function is given by:
\[
f'(x) = 5x^2(x^2 - 3)
\]
3. The second derivative of the function is given by:
\[
f''(x) = 10x(2x^2 - 3)
\]
4. The function passes through the point:
\[
f(0) = 0
\]
### Analysis of Given Conditions:
- **First Derivative \( f'(x) \)**:
- Determines the slope of the function \( f(x) \).
- Critical points are found by setting \( f'(x) = 0 \):
\[
5x^2(x^2 - 3) = 0
\]
Solutions: \( x = 0, \pm\sqrt{3} \)
- **Second Derivative \( f''(x) \)**:
- Determines the concavity of the function \( f(x) \).
- Possible inflection points are found by setting \( f''(x) = 0 \):
\[
10x(2x^2 - 3) = 0
\]
Solutions: \( x = 0, \pm\sqrt{\frac{3}{2}} \)
### Graphing Instructions:
- **Plot the Critical Points**:
- At \( x = 0, \pm\sqrt{3} \)
- **Determine Intervals of Increase](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F56488219-115f-4b7a-9fca-52b6982bead0%2F9871bd88-9b2a-4616-939e-6eabe900de7b%2Fils93z9_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Educational Resource: Graphing Functions with Specific Conditions
#### Task Description:
Sketch the graph of a single function \( f(x) \) that satisfies all of the following conditions, labeling all local extrema, inflection points, and asymptotes.
#### Steps to Follow:
1. **Identify Key Features**:
- **Local Extrema**: Points where the function changes from increasing to decreasing or vice versa.
- **Inflection Points**: Points where the concavity of the function changes.
- **Asymptotes**: Lines that the graph approaches but does not touch.
2. **Explicitly State the Intervals**:
- Intervals of increase and decrease.
- Intervals of concavity.
### Given Conditions:
1. **The domain of \( f(x) \)** is \( (-\infty, \infty) \).
2. The first derivative of the function is given by:
\[
f'(x) = 5x^2(x^2 - 3)
\]
3. The second derivative of the function is given by:
\[
f''(x) = 10x(2x^2 - 3)
\]
4. The function passes through the point:
\[
f(0) = 0
\]
### Analysis of Given Conditions:
- **First Derivative \( f'(x) \)**:
- Determines the slope of the function \( f(x) \).
- Critical points are found by setting \( f'(x) = 0 \):
\[
5x^2(x^2 - 3) = 0
\]
Solutions: \( x = 0, \pm\sqrt{3} \)
- **Second Derivative \( f''(x) \)**:
- Determines the concavity of the function \( f(x) \).
- Possible inflection points are found by setting \( f''(x) = 0 \):
\[
10x(2x^2 - 3) = 0
\]
Solutions: \( x = 0, \pm\sqrt{\frac{3}{2}} \)
### Graphing Instructions:
- **Plot the Critical Points**:
- At \( x = 0, \pm\sqrt{3} \)
- **Determine Intervals of Increase
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