Compute the discriminant D(x, y) of the function. f(x, y) = x³ + y² - 6x - 2y² + 2 (Express numbers in exact form. Use symbolic notation and fractions where needed.) D(x, y) =
Compute the discriminant D(x, y) of the function. f(x, y) = x³ + y² - 6x - 2y² + 2 (Express numbers in exact form. Use symbolic notation and fractions where needed.) D(x, y) =
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter3: Functions And Graphs
Section3.5: Graphs Of Functions
Problem 35E
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![### Problem Statement
Compute the discriminant \( D(x, y) \) of the function.
\[ f(x, y) = x^3 + y^4 - 6x - 2y^2 + 2 \]
(Express numbers in exact form. Use symbolic notation and fractions where needed.)
\[ D(x, y) = \]
### Solution Explanation
To compute the discriminant \( D(x, y) \) of the given function, we need to follow certain steps. The discriminant is often associated with the second-order partial derivatives of the function \( f(x, y) \).
Let's break down the steps:
1. Compute the first-order partial derivatives \( f_x \) and \( f_y \).
2. Compute the second-order partial derivatives \( f_{xx} \), \( f_{yy} \), and \( f_{xy} \).
3. Use these second-order partial derivatives to compute the discriminant \( D(x, y) \).
The general formula for the discriminant \( D(x, y) \) is:
\[ D(x, y) = f_{xx}f_{yy} - (f_{xy})^2 \]
Where \( f_{xx} \) and \( f_{yy} \) are the second-order partial derivatives with respect to \( x \) and \( y \) respectively, and \( f_{xy} \) is the mixed partial derivative.
Given the function:
\[ f(x, y) = x^3 + y^4 - 6x - 2y^2 + 2 \]
Let's compute the partial derivatives:
1. First-order partial derivatives:
- \( f_x = \frac{\partial}{\partial x} (x^3 + y^4 - 6x - 2y^2 + 2) = 3x^2 - 6 \)
- \( f_y = \frac{\partial}{\partial y} (x^3 + y^4 - 6x - 2y^2 + 2) = 4y^3 - 4y \)
2. Second-order partial derivatives and mixed partial derivative:
- \( f_{xx} = \frac{\partial}{\partial x}(3x^2 - 6) = 6x \)
- \( f_{yy} = \frac{\partial}{\partial y}(4y^3](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe132a23b-2bfe-4f64-9635-6f1845f8e4fa%2F7b57191e-9844-4906-a8c8-eb13f40c030e%2Fpjaeeh_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem Statement
Compute the discriminant \( D(x, y) \) of the function.
\[ f(x, y) = x^3 + y^4 - 6x - 2y^2 + 2 \]
(Express numbers in exact form. Use symbolic notation and fractions where needed.)
\[ D(x, y) = \]
### Solution Explanation
To compute the discriminant \( D(x, y) \) of the given function, we need to follow certain steps. The discriminant is often associated with the second-order partial derivatives of the function \( f(x, y) \).
Let's break down the steps:
1. Compute the first-order partial derivatives \( f_x \) and \( f_y \).
2. Compute the second-order partial derivatives \( f_{xx} \), \( f_{yy} \), and \( f_{xy} \).
3. Use these second-order partial derivatives to compute the discriminant \( D(x, y) \).
The general formula for the discriminant \( D(x, y) \) is:
\[ D(x, y) = f_{xx}f_{yy} - (f_{xy})^2 \]
Where \( f_{xx} \) and \( f_{yy} \) are the second-order partial derivatives with respect to \( x \) and \( y \) respectively, and \( f_{xy} \) is the mixed partial derivative.
Given the function:
\[ f(x, y) = x^3 + y^4 - 6x - 2y^2 + 2 \]
Let's compute the partial derivatives:
1. First-order partial derivatives:
- \( f_x = \frac{\partial}{\partial x} (x^3 + y^4 - 6x - 2y^2 + 2) = 3x^2 - 6 \)
- \( f_y = \frac{\partial}{\partial y} (x^3 + y^4 - 6x - 2y^2 + 2) = 4y^3 - 4y \)
2. Second-order partial derivatives and mixed partial derivative:
- \( f_{xx} = \frac{\partial}{\partial x}(3x^2 - 6) = 6x \)
- \( f_{yy} = \frac{\partial}{\partial y}(4y^3
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