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Consider the function f(x) = = - 3x²e-². c. Identify intervals of increase. 8

Consider the function f(x) = = - 3x²e-². c. Identify intervals of increase. 8

Advanced Engineering Mathematics
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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Question
## Analysis of the Function \( f(x) = -3x^2 e^{-\frac{x}{4}} \) ### Function Analysis Questions **c. Identify intervals of increase.** - The provided interval \( 8 < x < \infty \) is incorrect. **d. Identify intervals of decrease.** - The answer is left blank and needs to be determined. **e. Use parts a through d to identify any local maximums as ordered pairs.** - The answer box is left blank and points to areas that need to be analyzed using previous parts. **f. Use parts a through d to identify any local minimums as ordered pairs.** - The answer box is left blank and should be identified using previous analyses. ### Explanation This question involves finding where the function is increasing or decreasing and identifying any potential local maxima or minima based on given intervals and calculations of derivatives. To solve: 1. **Find the first derivative** of the function \( f(x) \). 2. **Determine critical points** by setting the derivative to zero. 3. **Test intervals** around critical points to determine where the function is increasing or decreasing. 4. **Identify local extrema** using the First or Second Derivative Test. This exercise is important for understanding the behavior of functions in calculus and can be applied in real-world contexts involving rate of change.
Transcribed Image Text:## Analysis of the Function \( f(x) = -3x^2 e^{-\frac{x}{4}} \) ### Function Analysis Questions **c. Identify intervals of increase.** - The provided interval \( 8 < x < \infty \) is incorrect. **d. Identify intervals of decrease.** - The answer is left blank and needs to be determined. **e. Use parts a through d to identify any local maximums as ordered pairs.** - The answer box is left blank and points to areas that need to be analyzed using previous parts. **f. Use parts a through d to identify any local minimums as ordered pairs.** - The answer box is left blank and should be identified using previous analyses. ### Explanation This question involves finding where the function is increasing or decreasing and identifying any potential local maxima or minima based on given intervals and calculations of derivatives. To solve: 1. **Find the first derivative** of the function \( f(x) \). 2. **Determine critical points** by setting the derivative to zero. 3. **Test intervals** around critical points to determine where the function is increasing or decreasing. 4. **Identify local extrema** using the First or Second Derivative Test. This exercise is important for understanding the behavior of functions in calculus and can be applied in real-world contexts involving rate of change.
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