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Given f(x, y) = -6x4 - 5xy² + 3y5, find fxx (x, y) = -72x2² fxy(x, y) = 10y ✓ X

Given f(x, y) = -6x4 - 5xy² + 3y5, find fxx (x, y) = -72x2² fxy(x, y) = 10y ✓ X

Calculus: Early Transcendentals
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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### Multivariable Calculus - Second Partial Derivatives #### Given: \[ f(x, y) = -6x^4 - 5xy^2 + 3y^5 \] ### Find: 1. The second partial derivative of \( f \) with respect to \( x \) twice, denoted as \( f_{xx}(x, y) \). 2. The mixed partial derivative of \( f \), where \( f \) is differentiated first with respect to \( x \) and then with respect to \( y \), denoted as \( f_{xy}(x, y) \). ### Solutions: #### Calculating \( f_{xx}(x, y) \): \[ f_{xx}(x, y) = -72x^2 \] Here, the answer is enclosed in a green box with a checkmark, indicating the correct solution. #### Calculating \( f_{xy}(x, y) \): \[ f_{xy}(x, y) = 10y \] However, this response is enclosed in a red box with an 'X', indicating that it is an incorrect solution. #### Explanation of the Results: - The second partial derivative \( f_{xx}(x, y) \) involves differentiating the original function \( f(x, y) \) twice with respect to \( x \). The correct result is \(-72x^2\), which is validated by the green checkmark. - The mixed partial derivative \( f_{xy}(x, y) \) is found by differentiating \( f(x, y) \) first with respect to \( x \) and then with respect to \( y \). The provided value \( 10y \) is incorrect, denoted by the red 'X'. You must re-evaluate the expression to find the correct mixed partial derivative.
Transcribed Image Text:### Multivariable Calculus - Second Partial Derivatives #### Given: \[ f(x, y) = -6x^4 - 5xy^2 + 3y^5 \] ### Find: 1. The second partial derivative of \( f \) with respect to \( x \) twice, denoted as \( f_{xx}(x, y) \). 2. The mixed partial derivative of \( f \), where \( f \) is differentiated first with respect to \( x \) and then with respect to \( y \), denoted as \( f_{xy}(x, y) \). ### Solutions: #### Calculating \( f_{xx}(x, y) \): \[ f_{xx}(x, y) = -72x^2 \] Here, the answer is enclosed in a green box with a checkmark, indicating the correct solution. #### Calculating \( f_{xy}(x, y) \): \[ f_{xy}(x, y) = 10y \] However, this response is enclosed in a red box with an 'X', indicating that it is an incorrect solution. #### Explanation of the Results: - The second partial derivative \( f_{xx}(x, y) \) involves differentiating the original function \( f(x, y) \) twice with respect to \( x \). The correct result is \(-72x^2\), which is validated by the green checkmark. - The mixed partial derivative \( f_{xy}(x, y) \) is found by differentiating \( f(x, y) \) first with respect to \( x \) and then with respect to \( y \). The provided value \( 10y \) is incorrect, denoted by the red 'X'. You must re-evaluate the expression to find the correct mixed partial derivative.
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