### Problem Statement**Objective:** Maximize the function $ f(x, y) = \sqrt{18 - x^2 - y^2} $.**Subject to the constraint:** $ x + y - 2 = 0 $.### ExplanationIn this problem, we are provided with a multivariable function $ f(x, y) $, which we need to maximize. The function is given by:\[ f(x, y) = \sqrt{18 - x^2 - y^2} \]Additionally, we have a constraint equation:\[ x + y - 2 = 0 \]This means that the values of $ x $ and $ y $ must satisfy this equation.### Details- $ f(x, y) $ represents a function with two variables $ x $ and $ y $.- The term $ \sqrt{18 - x^2 - y^2} $ suggests that the function is related to a circular region as the square root function ensures that the domain is limited by the condition $ 18 - x^2 - y^2 \geq 0 $.- The constraint $ x + y - 2 = 0 $ is a linear equation representing a line in the $ xy $-plane.To solve this problem, one might typically use optimization techniques such as the method of Lagrange multipliers, where the constraint is incorporated into the objective function to form a new function to be maximized or minimized.### Graphs/DiagramsThere are no specific graphs or diagrams provided in the problem. However, if visual aids were to be included:- **Graph of the Constraint Line $ x + y = 2 $**: A straight line in the $ xy $-plane, where for each $ x $, $ y $ is determined such that the sum of $ x $ and $ y $ equals 2. - **Graph of the Circular Boundary $ 18 - x^2 - y^2 = 0 $**: A circle centered at the origin with a radius of $ \sqrt{18} $ (since $ x^2 + y^2 = 18 $).Combining these, one could plot the circle and the line to find the feasible region and then determine the maximum value of the function within this region.

Name: ### Problem Statement**Objective:** Maximize the function $ f(x, y)
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Description: ### Problem Statement**Objective:** Maximize the function \( f(x, y) = \sqrt{18 - x^2 - y^2} $.**Subject to the constraint:** $ x + y - 2 = 0 $.### ExplanationIn this problem, we are provided with a multivariable function $ f(x, y) $, which we

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Answered: Maximize f(x, y) = 18 - x2 - y2, x + y…

Math

Calculus

Maximize f(x, y) = 18 - x2 - y2, x + y - 2 = 0.

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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Topic Video

Question

### Problem Statement

**Objective:** Maximize the function $ f(x, y) = \sqrt{18 - x^2 - y^2} $.

**Subject to the constraint:** $ x + y - 2 = 0 $.

### Explanation

In this problem, we are provided with a multivariable function $ f(x, y) $, which we need to maximize. The function is given by:

\[ f(x, y) = \sqrt{18 - x^2 - y^2} \]

Additionally, we have a constraint equation:

\[ x + y - 2 = 0 \]

This means that the values of $ x $ and $ y $ must satisfy this equation.

### Details

- $ f(x, y) $ represents a function with two variables $ x $ and $ y $.
- The term $ \sqrt{18 - x^2 - y^2} $ suggests that the function is related to a circular region as the square root function ensures that the domain is limited by the condition $ 18 - x^2 - y^2 \geq 0 $.
- The constraint $ x + y - 2 = 0 $ is a linear equation representing a line in the $ xy $-plane.

To solve this problem, one might typically use optimization techniques such as the method of Lagrange multipliers, where the constraint is incorporated into the objective function to form a new function to be maximized or minimized.

### Graphs/Diagrams

There are no specific graphs or diagrams provided in the problem. However, if visual aids were to be included:

- **Graph of the Constraint Line $ x + y = 2 $**: A straight line in the $ xy $-plane, where for each $ x $, $ y $ is determined such that the sum of $ x $ and $ y $ equals 2.

- **Graph of the Circular Boundary $ 18 - x^2 - y^2 = 0 $**: A circle centered at the origin with a radius of $ \sqrt{18} $ (since $ x^2 + y^2 = 18 $).

Combining these, one could plot the circle and the line to find the feasible region and then determine the maximum value of the function within this region.

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