**Sketch a graph of \( f(x) \) satisfying all the following conditions:** 1. \( f'(1) > 0 \) 2. \( f''(1) < 0 \) 3. \( f'(5) < 0 \) 4. \( f''(5) > 0 \) To better understand and satisfy these conditions, let’s break them down: 1. \( f'(1) > 0 \): - This means the slope of the function \( f(x) \) at \( x = 1 \) is positive. The graph of \( f(x) \) should be increasing at \( x = 1 \). 2. \( f''(1) < 0 \): - This indicates that the concavity of the function \( f(x) \) at \( x = 1 \) is negative. The graph of \( f(x) \) should be concave down at \( x = 1 \). 3. \( f'(5) < 0 \): - This means the slope of the function \( f(x) \) at \( x = 5 \) is negative. The graph of \( f(x) \) should be decreasing at \( x = 5 \). 4. \( f''(5) > 0 \): - This indicates that the concavity of the function \( f(x) \) at \( x = 5 \) is positive. The graph of \( f(x) \) should be concave up at \( x = 5 \). These conditions imply that at \( x = 1 \), the graph should be increasing and concave down, indicating a local maximum. At \( x = 5 \), the graph should be decreasing and concave up, indicating a local minimum. A possible graph might depict a local peak around \( x = 1 \) and a local valley around \( x = 5 \), smoothly transitioning between these points.
**Sketch a graph of \( f(x) \) satisfying all the following conditions:** 1. \( f'(1) > 0 \) 2. \( f''(1) < 0 \) 3. \( f'(5) < 0 \) 4. \( f''(5) > 0 \) To better understand and satisfy these conditions, let’s break them down: 1. \( f'(1) > 0 \): - This means the slope of the function \( f(x) \) at \( x = 1 \) is positive. The graph of \( f(x) \) should be increasing at \( x = 1 \). 2. \( f''(1) < 0 \): - This indicates that the concavity of the function \( f(x) \) at \( x = 1 \) is negative. The graph of \( f(x) \) should be concave down at \( x = 1 \). 3. \( f'(5) < 0 \): - This means the slope of the function \( f(x) \) at \( x = 5 \) is negative. The graph of \( f(x) \) should be decreasing at \( x = 5 \). 4. \( f''(5) > 0 \): - This indicates that the concavity of the function \( f(x) \) at \( x = 5 \) is positive. The graph of \( f(x) \) should be concave up at \( x = 5 \). These conditions imply that at \( x = 1 \), the graph should be increasing and concave down, indicating a local maximum. At \( x = 5 \), the graph should be decreasing and concave up, indicating a local minimum. A possible graph might depict a local peak around \( x = 1 \) and a local valley around \( x = 5 \), smoothly transitioning between these points.
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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Transcribed Image Text:**Sketch a graph of \( f(x) \) satisfying all the following conditions:**
1. \( f'(1) > 0 \)
2. \( f''(1) < 0 \)
3. \( f'(5) < 0 \)
4. \( f''(5) > 0 \)
To better understand and satisfy these conditions, let’s break them down:
1. \( f'(1) > 0 \):
- This means the slope of the function \( f(x) \) at \( x = 1 \) is positive. The graph of \( f(x) \) should be increasing at \( x = 1 \).
2. \( f''(1) < 0 \):
- This indicates that the concavity of the function \( f(x) \) at \( x = 1 \) is negative. The graph of \( f(x) \) should be concave down at \( x = 1 \).
3. \( f'(5) < 0 \):
- This means the slope of the function \( f(x) \) at \( x = 5 \) is negative. The graph of \( f(x) \) should be decreasing at \( x = 5 \).
4. \( f''(5) > 0 \):
- This indicates that the concavity of the function \( f(x) \) at \( x = 5 \) is positive. The graph of \( f(x) \) should be concave up at \( x = 5 \).
These conditions imply that at \( x = 1 \), the graph should be increasing and concave down, indicating a local maximum. At \( x = 5 \), the graph should be decreasing and concave up, indicating a local minimum. A possible graph might depict a local peak around \( x = 1 \) and a local valley around \( x = 5 \), smoothly transitioning between these points.
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