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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![**Problem 9:**
Use the D-test to find all relative extreme values for the function:
\[ f(x, y) = 2x^2 + 3y^2 + 2xy + 4x - 8y + 10 \]
**Explanation:**
The D-test, also known as the second derivative test, is utilized to determine the nature of critical points for multivariable functions. By examining the second partial derivatives, we construct a determinant (often denoted as D) that helps determine if the critical points are local minima, maxima, or saddle points.
The function given is a quadratic function of two variables, \(x\) and \(y\), with mixed partial derivatives involved. To apply the D-test, first, we find the critical points by setting the first partial derivatives to zero. Then, calculate the second partial derivatives, and use them to form the Hessian matrix. The determinant of this matrix is analyzed to conclude the behavior of the function at the critical points.
This exercise guides learners through the process of applying the D-test, offering insights into optimization within multivariable calculus.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F78c00f8e-d71a-411e-aae1-c1d19bd9af20%2F60ed41bc-edf2-49a6-99c7-b14aeef9a74a%2Fgcq1t6_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 9:**
Use the D-test to find all relative extreme values for the function:
\[ f(x, y) = 2x^2 + 3y^2 + 2xy + 4x - 8y + 10 \]
**Explanation:**
The D-test, also known as the second derivative test, is utilized to determine the nature of critical points for multivariable functions. By examining the second partial derivatives, we construct a determinant (often denoted as D) that helps determine if the critical points are local minima, maxima, or saddle points.
The function given is a quadratic function of two variables, \(x\) and \(y\), with mixed partial derivatives involved. To apply the D-test, first, we find the critical points by setting the first partial derivatives to zero. Then, calculate the second partial derivatives, and use them to form the Hessian matrix. The determinant of this matrix is analyzed to conclude the behavior of the function at the critical points.
This exercise guides learners through the process of applying the D-test, offering insights into optimization within multivariable calculus.
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