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Use the first derivative test to determine the location of each local extremum and the value of the function at that extremum. 3) f(x) = 4xe-x 4) f(x)=(7-6x)4/3-1

Use the first derivative test to determine the location of each local extremum and the value of the function at that extremum. 3) f(x) = 4xe-x 4) f(x)=(7-6x)4/3-1

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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### First Derivative Test for Finding Local Extrema In this exercise, you are required to use the first derivative test to determine the location of each local extremum and the value of the function at that extremum. #### Problem 3 \( f(x) = 4xe^{-x} \) #### Problem 4 \( f(x) = (7 - 6x)^{4/3} - 1 \) To solve these problems, follow these steps: 1. Find the first derivative of the given function, \( f'(x) \). 2. Determine the critical points by setting \( f'(x) = 0 \) and solving for \( x \). 3. Use the first derivative test to determine whether each critical point is a local maximum, local minimum, or neither. 4. Evaluate the original function \( f(x) \) at each critical point to find the value of the function at those points. These steps aid in highlighting the behavior of the function around the critical points and identifying whether those points correspond to local maxima or minima.
Transcribed Image Text:### First Derivative Test for Finding Local Extrema In this exercise, you are required to use the first derivative test to determine the location of each local extremum and the value of the function at that extremum. #### Problem 3 \( f(x) = 4xe^{-x} \) #### Problem 4 \( f(x) = (7 - 6x)^{4/3} - 1 \) To solve these problems, follow these steps: 1. Find the first derivative of the given function, \( f'(x) \). 2. Determine the critical points by setting \( f'(x) = 0 \) and solving for \( x \). 3. Use the first derivative test to determine whether each critical point is a local maximum, local minimum, or neither. 4. Evaluate the original function \( f(x) \) at each critical point to find the value of the function at those points. These steps aid in highlighting the behavior of the function around the critical points and identifying whether those points correspond to local maxima or minima.
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