4. Sketch the graph of a continuous function f(x) with domain [0, 2] which is differentiable up to second order on (0, 2) and satisfies: (a) f'(1) = 0 (b) f(x) < 0 in (0, 1) and (1, 2)
4. Sketch the graph of a continuous function f(x) with domain [0, 2] which is differentiable up to second order on (0, 2) and satisfies: (a) f'(1) = 0 (b) f(x) < 0 in (0, 1) and (1, 2)
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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![**Activity: Graphing a Continuous Function**
**Objective:**
Sketch the graph of a continuous function \( f(x) \) with the domain \([0, 2]\) that is differentiable up to the second order on \((0, 2)\), subject to the following conditions:
- \( f'(1) = 0 \)
- \( f''(x) < 0 \) in \((0, 1)\) and \((1, 2)\)
**Graph Analysis:**
The graph provided consists of a coordinate plane with standard grid markings:
- The horizontal axis (x-axis) ranges from 0 to 8.
- The vertical axis (y-axis) spans from -1 to 3.
**Instructions:**
1. **Identify Critical Points:**
- At \( x = 1 \), \( f'(1) = 0 \), indicating a horizontal tangent (potential maximum or minimum).
2. **Determine Concavity:**
- Since \( f''(x) < 0 \) over the intervals \((0, 1)\) and \((1, 2)\), the graph is concave down within these intervals.
3. **Construct the Sketch:**
- Begin at \( x = 0 \) and draw a concave down curve until \( x = 1 \).
- At \( x = 1 \), the curve should flatten due to the horizontal tangent.
- Continue with a concave down segment from \( x = 1 \) to \( x = 2 \).
Use this information to draw a graph that represents these mathematical properties accurately on the grid.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F593e4a4c-a9fc-4231-bacd-fbc9affb1612%2F5d030fbe-a6b5-493a-ac41-8a2b7b09a2bc%2Fzv2eea_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Activity: Graphing a Continuous Function**
**Objective:**
Sketch the graph of a continuous function \( f(x) \) with the domain \([0, 2]\) that is differentiable up to the second order on \((0, 2)\), subject to the following conditions:
- \( f'(1) = 0 \)
- \( f''(x) < 0 \) in \((0, 1)\) and \((1, 2)\)
**Graph Analysis:**
The graph provided consists of a coordinate plane with standard grid markings:
- The horizontal axis (x-axis) ranges from 0 to 8.
- The vertical axis (y-axis) spans from -1 to 3.
**Instructions:**
1. **Identify Critical Points:**
- At \( x = 1 \), \( f'(1) = 0 \), indicating a horizontal tangent (potential maximum or minimum).
2. **Determine Concavity:**
- Since \( f''(x) < 0 \) over the intervals \((0, 1)\) and \((1, 2)\), the graph is concave down within these intervals.
3. **Construct the Sketch:**
- Begin at \( x = 0 \) and draw a concave down curve until \( x = 1 \).
- At \( x = 1 \), the curve should flatten due to the horizontal tangent.
- Continue with a concave down segment from \( x = 1 \) to \( x = 2 \).
Use this information to draw a graph that represents these mathematical properties accurately on the grid.
![**Task: Sketching a Graph**
Sketch the graph of a continuous function \( f(x) \) with domain \([0, 2]\) which is differentiable up to the second order on \((0, 2)\) (\(f'(x)\) and \(f''(x)\) both exist for all \( x \) in \((0, 2)\)) and satisfies:
(a) \( f'(1) = 0 \)
(b) \( f''(x) > 0 \) in \((0, 1)\) and \((1, 2)\)
**Graph Explanation:**
- The graph is plotted on a coordinate plane with \(x\)-axis ranging from 0 to 8 and \(y\)-axis from -1 to 3.
- The challenge is to sketch a curve that meets the given conditions:
- It should have a horizontal tangent at \(x = 1\) because \(f'(1) = 0\).
- The curve should be concave up in the intervals \((0, 1)\) and \((1, 2)\) because \(f''(x) > 0\).
Marking point \(x = 1\) on the graph and the behavior around it will help visualize the curve appropriately. Adjust the curve to appear concave upwards before and after this point, maintaining the smoothness required by the conditions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F593e4a4c-a9fc-4231-bacd-fbc9affb1612%2F5d030fbe-a6b5-493a-ac41-8a2b7b09a2bc%2Fxsya08j_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Task: Sketching a Graph**
Sketch the graph of a continuous function \( f(x) \) with domain \([0, 2]\) which is differentiable up to the second order on \((0, 2)\) (\(f'(x)\) and \(f''(x)\) both exist for all \( x \) in \((0, 2)\)) and satisfies:
(a) \( f'(1) = 0 \)
(b) \( f''(x) > 0 \) in \((0, 1)\) and \((1, 2)\)
**Graph Explanation:**
- The graph is plotted on a coordinate plane with \(x\)-axis ranging from 0 to 8 and \(y\)-axis from -1 to 3.
- The challenge is to sketch a curve that meets the given conditions:
- It should have a horizontal tangent at \(x = 1\) because \(f'(1) = 0\).
- The curve should be concave up in the intervals \((0, 1)\) and \((1, 2)\) because \(f''(x) > 0\).
Marking point \(x = 1\) on the graph and the behavior around it will help visualize the curve appropriately. Adjust the curve to appear concave upwards before and after this point, maintaining the smoothness required by the conditions.
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