--- ### Problem Statement **4.** Given to the right is the graph of a function \( f' \), the DERIVATIVE of some function \( f \). Among the four graphs below are the functions \( f \) and \( f'' \).  - **Graph (i):** A graph showing a curve with a positive slope starting from the top left, decreasing through the x-axis, and continuing with a negative slope. - **Graph (ii):** A graph with a wave-like curve, crossing the x-axis three times. - **Graph (iii):** A graph with an S-shaped curve moving from the bottom left to top right. - **Graph (iv):** A graph of a curve that first decreases to a minimum point and then increases. **Task:** (a) State which of the four graphs above is the graph of \( f \). Clearly state your answer, and **explain** using complete sentences how you know your answer is right. (b) State which of the four graphs above is the graph of \( f'' \). Clearly state your answer, and **explain** using complete sentences how you know your answer is right. --- ### Explanation - The graph labeled nearby is a simple curve with a peak and a trough, which is indicative of a derivative graph. The task is to match this with the primary function \( f \) and the second derivative function \( f'' \) from the provided graphs. - **Graph Analysis** - **Graph (i):** Indicates change from an increasing function to a decreasing function. - **Graph (ii):** Illustrates possible inflection points due to multiple crossings. - **Graph (iii):** S-shaped graph suggesting continuous increase or decrease in the rate of change. - **Graph (iv):** A parabolic shape that could indicate a primary or secondary derivative nature due to symmetrical curvature. The objective is to correctly pair these descriptions with \( f \) and \( f'' \) considering the provided derivative. ---
--- ### Problem Statement **4.** Given to the right is the graph of a function \( f' \), the DERIVATIVE of some function \( f \). Among the four graphs below are the functions \( f \) and \( f'' \).  - **Graph (i):** A graph showing a curve with a positive slope starting from the top left, decreasing through the x-axis, and continuing with a negative slope. - **Graph (ii):** A graph with a wave-like curve, crossing the x-axis three times. - **Graph (iii):** A graph with an S-shaped curve moving from the bottom left to top right. - **Graph (iv):** A graph of a curve that first decreases to a minimum point and then increases. **Task:** (a) State which of the four graphs above is the graph of \( f \). Clearly state your answer, and **explain** using complete sentences how you know your answer is right. (b) State which of the four graphs above is the graph of \( f'' \). Clearly state your answer, and **explain** using complete sentences how you know your answer is right. --- ### Explanation - The graph labeled nearby is a simple curve with a peak and a trough, which is indicative of a derivative graph. The task is to match this with the primary function \( f \) and the second derivative function \( f'' \) from the provided graphs. - **Graph Analysis** - **Graph (i):** Indicates change from an increasing function to a decreasing function. - **Graph (ii):** Illustrates possible inflection points due to multiple crossings. - **Graph (iii):** S-shaped graph suggesting continuous increase or decrease in the rate of change. - **Graph (iv):** A parabolic shape that could indicate a primary or secondary derivative nature due to symmetrical curvature. The objective is to correctly pair these descriptions with \( f \) and \( f'' \) considering the provided derivative. ---
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
- **Graph (i):** A graph showing a curve with a positive slope starting from the top left, decreasing through the x-axis, and continuing with a negative slope.
- **Graph (ii):** A graph with a wave-like curve, crossing the x-axis three times.
- **Graph (iii):** A graph with an S-shaped curve moving from the bottom left to top right.
- **Graph (iv):** A graph of a curve that first decreases to a minimum point and then increases.
**Task:**
(a) State which of the four graphs above is the graph of \( f \). Clearly state your answer, and **explain** using complete sentences how you know your answer is right.
(b) State which of the four graphs above is the graph of \( f'' \). Clearly state your answer, and **explain** using complete sentences how you know your answer is right.
---
### Explanation
- The graph labeled nearby is a simple curve with a peak and a trough, which is indicative of a derivative graph. The task is to match this with the primary function \( f \) and the second derivative function \( f'' \) from the provided graphs.
- **Graph Analysis**
- **Graph (i):** Indicates change from an increasing function to a decreasing function.
- **Graph (ii):** Illustrates possible inflection points due to multiple crossings.
- **Graph (iii):** S-shaped graph suggesting continuous increase or decrease in the rate of change.
- **Graph (iv):** A parabolic shape that could indicate a primary or secondary derivative nature due to symmetrical curvature.
The objective is to correctly pair these descriptions with \( f \) and \( f'' \) considering the provided derivative.
---](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0752ec0c-8830-498b-90c1-b31442d20efd%2F1b9cea37-85b1-4d72-a5c1-04d20d46646f%2Fvcow5ap.jpeg&w=3840&q=75)
Transcribed Image Text:---
### Problem Statement
**4.** Given to the right is the graph of a function \( f' \), the DERIVATIVE of some function \( f \). Among the four graphs below are the functions \( f \) and \( f'' \).
- **Graph (i):** A graph showing a curve with a positive slope starting from the top left, decreasing through the x-axis, and continuing with a negative slope.
- **Graph (ii):** A graph with a wave-like curve, crossing the x-axis three times.
- **Graph (iii):** A graph with an S-shaped curve moving from the bottom left to top right.
- **Graph (iv):** A graph of a curve that first decreases to a minimum point and then increases.
**Task:**
(a) State which of the four graphs above is the graph of \( f \). Clearly state your answer, and **explain** using complete sentences how you know your answer is right.
(b) State which of the four graphs above is the graph of \( f'' \). Clearly state your answer, and **explain** using complete sentences how you know your answer is right.
---
### Explanation
- The graph labeled nearby is a simple curve with a peak and a trough, which is indicative of a derivative graph. The task is to match this with the primary function \( f \) and the second derivative function \( f'' \) from the provided graphs.
- **Graph Analysis**
- **Graph (i):** Indicates change from an increasing function to a decreasing function.
- **Graph (ii):** Illustrates possible inflection points due to multiple crossings.
- **Graph (iii):** S-shaped graph suggesting continuous increase or decrease in the rate of change.
- **Graph (iv):** A parabolic shape that could indicate a primary or secondary derivative nature due to symmetrical curvature.
The objective is to correctly pair these descriptions with \( f \) and \( f'' \) considering the provided derivative.
---
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