Sketch the graph of a function f that is continuous on [1, 5] and has the given properties. absolute maximum at 2, absolute minimum at 5, 4 is a critical number but there is no local maximum or minimum there
Sketch the graph of a function f that is continuous on [1, 5] and has the given properties. absolute maximum at 2, absolute minimum at 5, 4 is a critical number but there is no local maximum or minimum there
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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![**Title: Sketching Graphs in Calculus**
**Objective: Understanding and Sketching the Graph of a Continuous Function**
---
**Task Description:**
Sketch the graph of a function \( f \) that is continuous on the interval \([1, 5]\) and adheres to the following given properties:
- The function has an absolute maximum at \( x = 2 \).
- The function has an absolute minimum at \( x = 5 \).
- The point \( x = 4 \) is a critical number, but there is no local maximum or minimum at this point.
**Guidelines for Sketching the Graph:**
1. **Continuity on \([1, 5]\):**
The graph should have no breaks, jumps, or holes in the interval from \( x = 1 \) to \( x = 5 \).
2. **Absolute Maximum:**
At \( x = 2 \), the function reaches its highest value over the interval \([1, 5]\). The graph should attain a peak at this point.
3. **Absolute Minimum:**
At \( x = 5 \), the function reaches its lowest value over the interval \([1, 5]\). The graph should drop to its lowest point here.
4. **Critical Point at \( x = 4 \):**
The derivative of the function (slope of the tangent line) at \( x = 4 \) is zero, indicating a horizontal tangent. However, this point is neither a local maximum nor a local minimum, thus it might be a point of inflection or just a flat spot on the graph.
**Example Sketch Explanation:**
- **From \( x = 1 \) to \( x = 2 \):**
The graph should increase and reach its highest point at \( x = 2 \).
- **At \( x = 2 \):**
The graph peaks, representing the absolute maximum.
- **From \( x = 2 \) to \( x = 4 \):**
The graph should decrease towards \( x = 4 \).
- **At \( x = 4 \):**
The graph should flatten out momentarily, reflecting a critical point with no local max/min.
- **From \( x = 4 \) to \( x = 5 \):**
The graph should descend further and reach the lowest point at \( x =](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc1c90340-a748-4160-8298-b9e689632892%2F6f369489-6dad-46c4-a0e1-33485e4bf9ac%2F56w3cl_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Sketching Graphs in Calculus**
**Objective: Understanding and Sketching the Graph of a Continuous Function**
---
**Task Description:**
Sketch the graph of a function \( f \) that is continuous on the interval \([1, 5]\) and adheres to the following given properties:
- The function has an absolute maximum at \( x = 2 \).
- The function has an absolute minimum at \( x = 5 \).
- The point \( x = 4 \) is a critical number, but there is no local maximum or minimum at this point.
**Guidelines for Sketching the Graph:**
1. **Continuity on \([1, 5]\):**
The graph should have no breaks, jumps, or holes in the interval from \( x = 1 \) to \( x = 5 \).
2. **Absolute Maximum:**
At \( x = 2 \), the function reaches its highest value over the interval \([1, 5]\). The graph should attain a peak at this point.
3. **Absolute Minimum:**
At \( x = 5 \), the function reaches its lowest value over the interval \([1, 5]\). The graph should drop to its lowest point here.
4. **Critical Point at \( x = 4 \):**
The derivative of the function (slope of the tangent line) at \( x = 4 \) is zero, indicating a horizontal tangent. However, this point is neither a local maximum nor a local minimum, thus it might be a point of inflection or just a flat spot on the graph.
**Example Sketch Explanation:**
- **From \( x = 1 \) to \( x = 2 \):**
The graph should increase and reach its highest point at \( x = 2 \).
- **At \( x = 2 \):**
The graph peaks, representing the absolute maximum.
- **From \( x = 2 \) to \( x = 4 \):**
The graph should decrease towards \( x = 4 \).
- **At \( x = 4 \):**
The graph should flatten out momentarily, reflecting a critical point with no local max/min.
- **From \( x = 4 \) to \( x = 5 \):**
The graph should descend further and reach the lowest point at \( x =
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