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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![### Determining Extreme Values
**Problem Statement:**
Find the extreme values of \( f \) in the region described by the inequality.
The function to be analyzed is:
\[ f(x, y) = x^2 + y^2 + 6x - 6y \]
The region is defined by the inequality:
\[ x^2 + y^2 \leq 49 \]
**Solution:**
**Maximum Value:**
The maximum value of \( f \) in the given region is provided as:
\[ 49 + 42\sqrt{2} \]
This suggests that within the boundary \( x^2 + y^2 \leq 49 \), the function \( f \) attains its highest value at \( 49 + 42\sqrt{2} \).
**Minimum Value:**
No solution for the minimum value has been provided in the image.
**Explanation of the Region:**
The region described by the inequality \( x^2 + y^2 \leq 49 \) represents a circle with a radius of 7 centered at the origin \((0, 0)\).
**Understanding the Context:**
The given function \( f(x, y) = x^2 + y^2 + 6x - 6y \) is a quadratic function representing a paraboloid when plotted. The task involves finding the extreme (maximum and minimum) values of this function within the circular boundary defined above.
Such problems commonly involve using techniques from calculus, such as Lagrange multipliers, or geometric insights to determine the points at which these extreme values occur. It's important to evaluate both the interior and the boundary of the region to ensure capturing all potential extreme values.
This example outlines that the maximum value has been identified, but further work is needed to find the minimum value, typically requiring optimization methods and critical point analysis.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe6ce2240-b3fe-4331-98ee-3937416f40f5%2F93205cce-4e61-4de2-9cc1-6293417fb2a1%2Fvqmv2sw_processed.png&w=3840&q=75)
Transcribed Image Text:### Determining Extreme Values
**Problem Statement:**
Find the extreme values of \( f \) in the region described by the inequality.
The function to be analyzed is:
\[ f(x, y) = x^2 + y^2 + 6x - 6y \]
The region is defined by the inequality:
\[ x^2 + y^2 \leq 49 \]
**Solution:**
**Maximum Value:**
The maximum value of \( f \) in the given region is provided as:
\[ 49 + 42\sqrt{2} \]
This suggests that within the boundary \( x^2 + y^2 \leq 49 \), the function \( f \) attains its highest value at \( 49 + 42\sqrt{2} \).
**Minimum Value:**
No solution for the minimum value has been provided in the image.
**Explanation of the Region:**
The region described by the inequality \( x^2 + y^2 \leq 49 \) represents a circle with a radius of 7 centered at the origin \((0, 0)\).
**Understanding the Context:**
The given function \( f(x, y) = x^2 + y^2 + 6x - 6y \) is a quadratic function representing a paraboloid when plotted. The task involves finding the extreme (maximum and minimum) values of this function within the circular boundary defined above.
Such problems commonly involve using techniques from calculus, such as Lagrange multipliers, or geometric insights to determine the points at which these extreme values occur. It's important to evaluate both the interior and the boundary of the region to ensure capturing all potential extreme values.
This example outlines that the maximum value has been identified, but further work is needed to find the minimum value, typically requiring optimization methods and critical point analysis.
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