Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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![### Understanding Increasing Functions
#### Problem Statement:
Given a function \( s(x) \) such that \( s'(x) > 1 \) on the interval \([0, 7]\). If \( s(0) = -9 \), which of the following could be the value of \( s(7) \)? Select all correct values.
#### Options:
- \[ \quad \] \(-17\)
- \[ \quad \] \(-8\)
- \[ \quad \] \(12\)
- \[ \quad \] \(15\)
- \[ \quad \] \(21\)
#### Explanation:
1. **Understanding the Condition \( s'(x) > 1 \):**
- This implies that the derivative of \( s(x) \) is always greater than 1 on the interval \([0, 7]\).
- A derivative greater than 1 means the function \( s(x) \) is increasing at a rate faster than a linear function with a slope of 1.
2. **Calculating the Minimum Increase:**
- Since the rate at which \( s(x) \) increases is always greater than 1, the minimum increase in \( s(x) \) over the interval \([0, 7]\) can be calculated as:
\[
s(7) - s(0) > 7 \cdot 1 = 7
\]
- Given \( s(0) = -9 \):
\[
s(7) > -9 + 7 = -2
\]
- Therefore, \( s(7) \) must be greater than \(-2\).
3. **Evaluating the Options:**
- \(-17\) is not greater than \(-2\).
- \(-8\) is not greater than \(-2\).
- \(12\) is greater than \(-2\).
- \(15\) is greater than \(-2\).
- \(21\) is greater than \(-2\).
Based on this evaluation, the correct values for \( s(7) \) are:
- \[ \quad \] \(12\)
- \[ \quad \] \(15\)
- \[ \quad \] \(21\)
Feel free to select one or](https://content.bartleby.com/qna-images/question/b47bac88-cba5-40e5-9f3f-db7316eb95a3/0ff77223-ae0b-4fc7-8615-ca816aa3c8bb/jilsll_processed.jpeg)
Transcribed Image Text:### Understanding Increasing Functions
#### Problem Statement:
Given a function \( s(x) \) such that \( s'(x) > 1 \) on the interval \([0, 7]\). If \( s(0) = -9 \), which of the following could be the value of \( s(7) \)? Select all correct values.
#### Options:
- \[ \quad \] \(-17\)
- \[ \quad \] \(-8\)
- \[ \quad \] \(12\)
- \[ \quad \] \(15\)
- \[ \quad \] \(21\)
#### Explanation:
1. **Understanding the Condition \( s'(x) > 1 \):**
- This implies that the derivative of \( s(x) \) is always greater than 1 on the interval \([0, 7]\).
- A derivative greater than 1 means the function \( s(x) \) is increasing at a rate faster than a linear function with a slope of 1.
2. **Calculating the Minimum Increase:**
- Since the rate at which \( s(x) \) increases is always greater than 1, the minimum increase in \( s(x) \) over the interval \([0, 7]\) can be calculated as:
\[
s(7) - s(0) > 7 \cdot 1 = 7
\]
- Given \( s(0) = -9 \):
\[
s(7) > -9 + 7 = -2
\]
- Therefore, \( s(7) \) must be greater than \(-2\).
3. **Evaluating the Options:**
- \(-17\) is not greater than \(-2\).
- \(-8\) is not greater than \(-2\).
- \(12\) is greater than \(-2\).
- \(15\) is greater than \(-2\).
- \(21\) is greater than \(-2\).
Based on this evaluation, the correct values for \( s(7) \) are:
- \[ \quad \] \(12\)
- \[ \quad \] \(15\)
- \[ \quad \] \(21\)
Feel free to select one or
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