**Transcription and Explanation for Educational Purposes****Text:**Pictured are the contours (z=0 to z=4.5 contour interval 0.5) of\[ f(x, y) = x^2 + 2xy + y^2 \]together with the domain\[ D = \{ (x, y) \in \mathbb{R}^2 : 0 \leq y \leq 2, \, y - 2 \leq x \leq 1 - \frac{y}{2} \} \].Explain how the graph can be used to determine the global maximum and the global minimum of the function on $ D $. Verify by including a table of values for any critical points (in $ D $) and boundary-critical points of the function.**Diagram Explanation:**The diagram includes a contour plot of the function $ f(x, y) = x^2 + 2xy + y^2 $. The domain $ D $ is a triangular region highlighted in blue, bounded by the lines $ y = 0 $, $ y = 2 $, $ x = y - 2 $, and $ x = 1 - \frac{y}{2} $. The contour lines represent levels of the function $ f(x, y) $, spaced at intervals of 0.5 from $ z = 0 $ to $ z = 4.5 $.Each contour line connects points of equal function value. As you observe the contour lines moving through the domain $ D $, the density and spacing can help identify regions where the function increases or decreases. Tighter contour lines indicate a steeper gradient.**Analysis Guidance:**To determine the global maximum and minimum of $ f(x, y) $ on the domain $ D $, you can:1. **Evaluate Critical Points:** Calculate the gradient of $ f $ and set it equal to zero to find critical points within the domain.2. **Check Boundary Points:** Evaluate the function along the boundaries of $ D $ to identify any potential maxima or minima.3. **Compare Values:** Compile a table of function values at the critical points and along the boundary to determine the absolute maximum and minimum.This analysis allows you to explore and explain the behavior of the function within the specified domain, using both algebraic and graphical approaches.

Answered: Pictured are the contours (z=0 to z=4.5…

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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Question

**Transcription and Explanation for Educational Purposes**

**Text:**

Pictured are the contours (z=0 to z=4.5 contour interval 0.5) of
\[ f(x, y) = x^2 + 2xy + y^2 \]
together with the domain
\[ D = \{ (x, y) \in \mathbb{R}^2 : 0 \leq y \leq 2, \, y - 2 \leq x \leq 1 - \frac{y}{2} \} \].
Explain how the graph can be used to determine the global maximum and the global minimum of the function on $ D $. Verify by including a table of values for any critical points (in $ D $) and boundary-critical points of the function.

**Diagram Explanation:**

The diagram includes a contour plot of the function $ f(x, y) = x^2 + 2xy + y^2 $. The domain $ D $ is a triangular region highlighted in blue, bounded by the lines $ y = 0 $, $ y = 2 $, $ x = y - 2 $, and $ x = 1 - \frac{y}{2} $. The contour lines represent levels of the function $ f(x, y) $, spaced at intervals of 0.5 from $ z = 0 $ to $ z = 4.5 $.

Each contour line connects points of equal function value. As you observe the contour lines moving through the domain $ D $, the density and spacing can help identify regions where the function increases or decreases. Tighter contour lines indicate a steeper gradient.

**Analysis Guidance:**

To determine the global maximum and minimum of $ f(x, y) $ on the domain $ D $, you can:

1. **Evaluate Critical Points:** Calculate the gradient of $ f $ and set it equal to zero to find critical points within the domain.
2. **Check Boundary Points:** Evaluate the function along the boundaries of $ D $ to identify any potential maxima or minima.
3. **Compare Values:** Compile a table of function values at the critical points and along the boundary to determine the absolute maximum and minimum.

This analysis allows you to explore and explain the behavior of the function within the specified domain, using both algebraic and graphical approaches.

Expert Solution

Step 1: Writing down the given information

Given the contours from $z = 0$ to $z = 4.5$ .

$f (x, y) = z = x^{2} + 2 x y + y^{2}$ is the surface with domain $D = {(x, y) \in R^{2} | y \in [0, 2], x \in [y - 2, 1 - \frac{y}{2}]}$ .

Let us analyze the contour lines for $z = 0$ and $z = 4$ .

For $z = 0$ , the contour line will be

$x squared plus 2 x y plus y squared equals 0 rightwards double arrow left parenthesis x plus y right parenthesis squared equals 0 rightwards double arrow y equals negative x.$

For $z = 4,$ the contour lines will be

$x squared plus 2 x y plus y squared equals 4 open parentheses x plus y close parentheses squared equals 4 x plus y equals plus-or-minus 2 y equals negative x plus-or-minus 2.$