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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I do not understand how to parameterize the function using critical points.
![**Problem Statement:**
Determine the minimum and maximum values of the function \( f(x, y) = x^2 + 2x - 2y \) over the region defined by:
\[ D = \{ (x, y) : x^2 \leq y \leq 4, -2 \leq x \leq 2 \} \]
**Instructions:**
1. **Understand the Function:** The function \( f(x, y) \) is a quadratic in terms of \( x \) and linear in terms of \( y \).
2. **Analyze the Set \( D \):**
- The inequality \( x^2 \leq y \) describes a region above the parabola \( y = x^2 \).
- The inequality \( y \leq 4 \) restricts the region to lie below the line \( y = 4 \).
- The boundaries for \( x \) are \(-2 \leq x \leq 2\), which means the region is also constrained horizontally between \( x = -2 \) and \( x = 2 \).
3. **Graphical Representation:**
- To visualize the region \( D \), sketch the parabola \( y = x^2 \) and the horizontal line \( y = 4 \).
- Shade the area that lies above the parabola and below the line.
- The x-bounds \(-2\) and \(2\) create vertical lines that further narrow the shaded region.
4. **Finding Extrema:**
- To find the extrema, you may evaluate the function at critical points within the region, as well as along the boundaries \( y = x^2 \) and \( y = 4 \).
- Check endpoints and boundary intersections by substituting coordinates into \( f(x, y) \).
**Conclusion:**
Conclude by identifying the coordinates of points where the function attains its minimum and maximum values, specifying these values explicitly.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffa923b6f-81dd-482c-8885-6de6bc295751%2F3d237e6a-da82-42ec-a847-75ca2ccdd26d%2Fwxoraf3_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Determine the minimum and maximum values of the function \( f(x, y) = x^2 + 2x - 2y \) over the region defined by:
\[ D = \{ (x, y) : x^2 \leq y \leq 4, -2 \leq x \leq 2 \} \]
**Instructions:**
1. **Understand the Function:** The function \( f(x, y) \) is a quadratic in terms of \( x \) and linear in terms of \( y \).
2. **Analyze the Set \( D \):**
- The inequality \( x^2 \leq y \) describes a region above the parabola \( y = x^2 \).
- The inequality \( y \leq 4 \) restricts the region to lie below the line \( y = 4 \).
- The boundaries for \( x \) are \(-2 \leq x \leq 2\), which means the region is also constrained horizontally between \( x = -2 \) and \( x = 2 \).
3. **Graphical Representation:**
- To visualize the region \( D \), sketch the parabola \( y = x^2 \) and the horizontal line \( y = 4 \).
- Shade the area that lies above the parabola and below the line.
- The x-bounds \(-2\) and \(2\) create vertical lines that further narrow the shaded region.
4. **Finding Extrema:**
- To find the extrema, you may evaluate the function at critical points within the region, as well as along the boundaries \( y = x^2 \) and \( y = 4 \).
- Check endpoints and boundary intersections by substituting coordinates into \( f(x, y) \).
**Conclusion:**
Conclude by identifying the coordinates of points where the function attains its minimum and maximum values, specifying these values explicitly.
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Maximum and Minimum over a Region.
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