Chapter3: Functions
Section3.5: Transformation Of Functions
Problem 5SE: How can you determine whether a function is odd or even from the formula of the function?
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![**Problem Statement:**
Find the absolute maximum value of the function.
\[ f(x) = 2(x - e^x) \]
**Detailed Solution:**
1. **Identify the Function**: The function given is \( f(x) = 2(x - e^x) \).
2. **Find the Critical Points**:
- To find the maximum value, we start by finding the derivative of the function and setting it to zero.
\[ f'(x) = \frac{d}{dx}[2(x - e^x)] \]
\[ f'(x) = 2(1 - e^x) \]
- Set the derivative equal to zero to find the critical points.
\[ 2(1 - e^x) = 0 \]
\[ 1 - e^x = 0 \]
\[ e^x = 1 \]
\[ x = \ln(1) \]
\[ x = 0 \]
3. **Evaluate the Function at the Critical Points**:
- Calculate \( f(x) \) at \( x = 0 \):
\[ f(0) = 2(0 - e^0) \]
\[ f(0) = 2(0 - 1) \]
\[ f(0) = -2 \]
4. **Check End Behavior**:
- As \( x \to -\infty \):
\[ f(x) \to -\infty \]
- As \( x \to \infty \):
\[ f(x) \to -\infty \]
5. **Conclusion**: The absolute maximum value of \( f(x) = 2(x - e^x) \) occurs at \( x = 0 \) and the maximum value is \( f(0) = -2 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd1148f2e-12e2-4d7a-aa19-e54736499ad8%2Fde2459a2-cb36-4f54-a811-bb2fc07e079b%2Fg6d7yn_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the absolute maximum value of the function.
\[ f(x) = 2(x - e^x) \]
**Detailed Solution:**
1. **Identify the Function**: The function given is \( f(x) = 2(x - e^x) \).
2. **Find the Critical Points**:
- To find the maximum value, we start by finding the derivative of the function and setting it to zero.
\[ f'(x) = \frac{d}{dx}[2(x - e^x)] \]
\[ f'(x) = 2(1 - e^x) \]
- Set the derivative equal to zero to find the critical points.
\[ 2(1 - e^x) = 0 \]
\[ 1 - e^x = 0 \]
\[ e^x = 1 \]
\[ x = \ln(1) \]
\[ x = 0 \]
3. **Evaluate the Function at the Critical Points**:
- Calculate \( f(x) \) at \( x = 0 \):
\[ f(0) = 2(0 - e^0) \]
\[ f(0) = 2(0 - 1) \]
\[ f(0) = -2 \]
4. **Check End Behavior**:
- As \( x \to -\infty \):
\[ f(x) \to -\infty \]
- As \( x \to \infty \):
\[ f(x) \to -\infty \]
5. **Conclusion**: The absolute maximum value of \( f(x) = 2(x - e^x) \) occurs at \( x = 0 \) and the maximum value is \( f(0) = -2 \).
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