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Q3. (0.75 Point) Determine where V(z)=z4(2z - 8)³ is increasing and decreasing.

Q3. (0.75 Point) Determine where V(z)=z4(2z - 8)³ is increasing and decreasing.

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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### Calculus Question **Q3. (0.75 Point):** Determine where \( V(z) = z^4 (2z - 8)^3 \) is increasing and decreasing. --- To solve this problem, follow these steps: 1. **Find the First Derivative**: The first derivative of \( V(z) \) with respect to \( z \), denoted as \( V'(z) \), will help in determining the critical points. 2. **Set the First Derivative Equal to Zero**: Solve \( V'(z) = 0 \) to find the critical points. 3. **Determine the Sign of the First Derivative**: Analyze the intervals around the critical points to determine where the function is increasing (where \( V'(z) > 0 \)) and where it is decreasing (where \( V'(z) < 0 \)). 4. **Conclusion**: Based on the sign of the first derivative in the intervals, determine the regions where \( V(z) \) is increasing or decreasing. ### Explanation of the Function - The function given is a product of \( z^4 \) and \( (2z - 8)^3 \). - \( z^4 \) is always non-negative since it is an even power. - \( (2z - 8)^3 \) determines the behavior of the function around \( z = 4 \). ### Analysis Consider critical points and intervals, and apply the first derivative test to conclude on increasing and decreasing intervals. --- By working through these steps, one can fully characterize the behavior of the function \( V(z) \) with respect to increasing and decreasing intervals.
Transcribed Image Text:### Calculus Question **Q3. (0.75 Point):** Determine where \( V(z) = z^4 (2z - 8)^3 \) is increasing and decreasing. --- To solve this problem, follow these steps: 1. **Find the First Derivative**: The first derivative of \( V(z) \) with respect to \( z \), denoted as \( V'(z) \), will help in determining the critical points. 2. **Set the First Derivative Equal to Zero**: Solve \( V'(z) = 0 \) to find the critical points. 3. **Determine the Sign of the First Derivative**: Analyze the intervals around the critical points to determine where the function is increasing (where \( V'(z) > 0 \)) and where it is decreasing (where \( V'(z) < 0 \)). 4. **Conclusion**: Based on the sign of the first derivative in the intervals, determine the regions where \( V(z) \) is increasing or decreasing. ### Explanation of the Function - The function given is a product of \( z^4 \) and \( (2z - 8)^3 \). - \( z^4 \) is always non-negative since it is an even power. - \( (2z - 8)^3 \) determines the behavior of the function around \( z = 4 \). ### Analysis Consider critical points and intervals, and apply the first derivative test to conclude on increasing and decreasing intervals. --- By working through these steps, one can fully characterize the behavior of the function \( V(z) \) with respect to increasing and decreasing intervals.
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