18. Let f(x)=2- |2x - 1. Show that there is no value of e such that f(3) -f(0) = f'(c)(3-0). Why does this not contradict the Mean Value Theorem?
18. Let f(x)=2- |2x - 1. Show that there is no value of e such that f(3) -f(0) = f'(c)(3-0). Why does this not contradict the Mean Value Theorem?
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Problem 18:**
Let \( f(x) = 2 - |2x - 1| \). Show that there is no value of \( c \) such that
\[
\frac{f(3) - f(0)}{3 - 0} = f'(c)
\]
Why does this not contradict the Mean Value Theorem?
---
**Solution Explanation:**
To solve this problem, we need to first find the expression for \( f(x) \) and subsequently calculate \( f'(x) \).
1. **Evaluate \( f(3) \) and \( f(0) \):**
- Since \( f(x) = 2 - |2x - 1| \), let's consider the two cases for \( |2x - 1| \):
- If \( 2x - 1 \geq 0 \), then \( |2x - 1| = 2x - 1 \).
- If \( 2x - 1 < 0 \), then \( |2x - 1| = -(2x - 1) = -2x + 1 \).
- Calculate \( f(3) \):
\[
f(3) = 2 - |2(3) - 1| = 2 - |6 - 1| = 2 - 5 = -3
\]
- Calculate \( f(0) \):
\[
f(0) = 2 - |2(0) - 1| = 2 - |-1| = 2 - 1 = 1
\]
2. **Calculate the average rate of change from \( x = 0 \) to \( x = 3 \):**
\[
\frac{f(3) - f(0)}{3 - 0} = \frac{-3 - 1}{3} = \frac{-4}{3}
\]
3. **Calculate the derivative \( f'(x) \):**
- Use the piecewise definition for \( |2x - 1| \):
- For \( x \geq \frac{1}{2} \), \( f(x) = 2 - (2x - 1) = 3 - 2x \).
- Therefore, \( f'(x)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F90d2137b-83f2-4c15-8300-1a033b76054d%2F4c9983db-5133-45ad-ad03-2424d335b5a0%2Fesxn25u_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem 18:**
Let \( f(x) = 2 - |2x - 1| \). Show that there is no value of \( c \) such that
\[
\frac{f(3) - f(0)}{3 - 0} = f'(c)
\]
Why does this not contradict the Mean Value Theorem?
---
**Solution Explanation:**
To solve this problem, we need to first find the expression for \( f(x) \) and subsequently calculate \( f'(x) \).
1. **Evaluate \( f(3) \) and \( f(0) \):**
- Since \( f(x) = 2 - |2x - 1| \), let's consider the two cases for \( |2x - 1| \):
- If \( 2x - 1 \geq 0 \), then \( |2x - 1| = 2x - 1 \).
- If \( 2x - 1 < 0 \), then \( |2x - 1| = -(2x - 1) = -2x + 1 \).
- Calculate \( f(3) \):
\[
f(3) = 2 - |2(3) - 1| = 2 - |6 - 1| = 2 - 5 = -3
\]
- Calculate \( f(0) \):
\[
f(0) = 2 - |2(0) - 1| = 2 - |-1| = 2 - 1 = 1
\]
2. **Calculate the average rate of change from \( x = 0 \) to \( x = 3 \):**
\[
\frac{f(3) - f(0)}{3 - 0} = \frac{-3 - 1}{3} = \frac{-4}{3}
\]
3. **Calculate the derivative \( f'(x) \):**
- Use the piecewise definition for \( |2x - 1| \):
- For \( x \geq \frac{1}{2} \), \( f(x) = 2 - (2x - 1) = 3 - 2x \).
- Therefore, \( f'(x)
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