Carefully sketch the graph of a continuous function f which satisfies the following properties:
Carefully sketch the graph of a continuous function f which satisfies the following properties:
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![**Problem Statement:**
14.) Carefully sketch the graph of a continuous function \( f \) which satisfies the following properties:
**Signs of \( f' \):**
- \( (-\infty, -2) \): \( f' < 0 \)
- \( (-2, 0) \): \( f' > 0 \)
- \( (0, \infty) \): \( f' < 0 \)
**Signs of \( f'' \):**
- \( (-\infty, -1) \): \( f'' > 0 \)
- \( (-1, 1) \): \( f'' < 0 \)
- \( (1, \infty) \): \( f'' > 0 \)
**Limits:**
- \( \lim_{{x \to -\infty}} f(x) = \infty \)
- \( \lim_{{x \to \infty}} f(x) = -1 \)
**Table of Values:**
\[
\begin{array}{c|c}
x & f(x) \\
\hline
-2 & -1 \\
-1 & 0 \\
0 & 2 \\
1 & 1 \\
\end{array}
\]
**Explanation:**
- **Signs of \( f' \):**
- The function \( f \) is decreasing for \( x < -2 \) and \( x > 0 \).
- It is increasing between \( x = -2 \) and \( x = 0 \).
- **Signs of \( f'' \):**
- The function \( f \) is concave up for \( x < -1 \) and \( x > 1 \).
- It is concave down between \( x = -1 \) and \( x = 1 \).
- **Behavior at Infinity:**
- \( f(x) \) approaches infinity as \( x \) approaches negative infinity.
- \( f(x) \) approaches -1 as \( x \) approaches positive infinity.
- **Table of Values:**
- At \( x = -2 \), \( f(x) = -1 \).
- At \( x = -1 \), \( f(x) = 0 \).
- At \( x = 0 \), \( f(x) = 2 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc2d98c66-c867-40dc-8395-8baf72422c58%2F08e03442-6fbe-4336-8de8-4f6ac38ff9c8%2F3fa0yo_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
14.) Carefully sketch the graph of a continuous function \( f \) which satisfies the following properties:
**Signs of \( f' \):**
- \( (-\infty, -2) \): \( f' < 0 \)
- \( (-2, 0) \): \( f' > 0 \)
- \( (0, \infty) \): \( f' < 0 \)
**Signs of \( f'' \):**
- \( (-\infty, -1) \): \( f'' > 0 \)
- \( (-1, 1) \): \( f'' < 0 \)
- \( (1, \infty) \): \( f'' > 0 \)
**Limits:**
- \( \lim_{{x \to -\infty}} f(x) = \infty \)
- \( \lim_{{x \to \infty}} f(x) = -1 \)
**Table of Values:**
\[
\begin{array}{c|c}
x & f(x) \\
\hline
-2 & -1 \\
-1 & 0 \\
0 & 2 \\
1 & 1 \\
\end{array}
\]
**Explanation:**
- **Signs of \( f' \):**
- The function \( f \) is decreasing for \( x < -2 \) and \( x > 0 \).
- It is increasing between \( x = -2 \) and \( x = 0 \).
- **Signs of \( f'' \):**
- The function \( f \) is concave up for \( x < -1 \) and \( x > 1 \).
- It is concave down between \( x = -1 \) and \( x = 1 \).
- **Behavior at Infinity:**
- \( f(x) \) approaches infinity as \( x \) approaches negative infinity.
- \( f(x) \) approaches -1 as \( x \) approaches positive infinity.
- **Table of Values:**
- At \( x = -2 \), \( f(x) = -1 \).
- At \( x = -1 \), \( f(x) = 0 \).
- At \( x = 0 \), \( f(x) = 2 \).
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 3 steps with 1 images
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
