the second pic is from the answer key, so the answers should match 6. Absolute maximum value is $ e^{1/4} $, absolute minimum value is $ e^{-1/4} $. ### Problem 6: Absolute Maximum and Minimum Values**Objective:**Find the absolute maximum and minimum values of the function $ f(x, y) = e^{-xy} $ within the region defined by $ D $.**Region Definition:**\[ D = \{(x, y) \mid x^2 + 4y^2 \leq 1\} \]This defines an elliptical region in the $ xy $-plane.**Function:**\[ f(x, y) = e^{-xy} \]To find the absolute maximum and minimum values of the given function $ f(x, y) $ within the region $ D $, one must evaluate the function over the entire region, including boundaries and interior points. The critical points inside the elliptical region, as well as the points on the boundary $ x^2 + 4y^2 = 1 $, should be considered.**Steps to Solve:**1. **Identify critical points** inside the region by setting up and solving the partial derivatives $ \frac{\partial f}{\partial x} $ and $ \frac{\partial f}{\partial y} $ equal to zero.2. **Evaluate the function on the boundary** of the region defined by $ x^2 + 4y^2 = 1 $. This can often be done using parameterization or Lagrange multipliers.3. Compare the values of the function at all identified points to determine the absolute maximum and minimum values.**Region Description:**The region $ D $ is an ellipse centered at the origin with a semi-major axis of 1 along the x-axis and a semi-minor axis of 0.5 along the y-axis. This is due to the equation $ x^2 + 4y^2 = 1 $, which is normalized to fit into the unit circle when plotted.**Application:**This problem highlights the method of finding extreme values of a function constrained within a specific region. It's useful in various fields, including optimization, economics, and engineering.

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6. Find the absolute maximum and minimum values of f(x, y) = e¯ region D= {(x, y) : x² + 4y² ≤ 1}. e-xy in the

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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the second pic is from the answer key, so the answers should match

6. Absolute maximum value is $ e^{1/4} $, absolute minimum value is $ e^{-1/4} $.

### Problem 6: Absolute Maximum and Minimum Values

**Objective:**
Find the absolute maximum and minimum values of the function $ f(x, y) = e^{-xy} $ within the region defined by $ D $.

**Region Definition:**
\[ D = \{(x, y) \mid x^2 + 4y^2 \leq 1\} \]

This defines an elliptical region in the $ xy $-plane.

**Function:**
\[ f(x, y) = e^{-xy} \]

To find the absolute maximum and minimum values of the given function $ f(x, y) $ within the region $ D $, one must evaluate the function over the entire region, including boundaries and interior points. The critical points inside the elliptical region, as well as the points on the boundary $ x^2 + 4y^2 = 1 $, should be considered.

**Steps to Solve:**
1. **Identify critical points** inside the region by setting up and solving the partial derivatives $ \frac{\partial f}{\partial x} $ and $ \frac{\partial f}{\partial y} $ equal to zero.
2. **Evaluate the function on the boundary** of the region defined by $ x^2 + 4y^2 = 1 $. This can often be done using parameterization or Lagrange multipliers.
3. Compare the values of the function at all identified points to determine the absolute maximum and minimum values.

**Region Description:**
The region $ D $ is an ellipse centered at the origin with a semi-major axis of 1 along the x-axis and a semi-minor axis of 0.5 along the y-axis. This is due to the equation $ x^2 + 4y^2 = 1 $, which is normalized to fit into the unit circle when plotted.

**Application:**
This problem highlights the method of finding extreme values of a function constrained within a specific region. It's useful in various fields, including optimization, economics, and engineering.

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