Consider the function f(x, y) = 2x² − y² + 4x + 6y subject to x + 2y = 20 a) Find the extreme point b) Determine the nature of the extreme point
Consider the function f(x, y) = 2x² − y² + 4x + 6y subject to x + 2y = 20 a) Find the extreme point b) Determine the nature of the extreme point
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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Please review the following for accuracy and please provide corrections where necessary:
Consider the function f(x, y) = 2x² − y² + 4x + 6y subject to x + 2y = 20
a) Find the extreme point
b) Determine the nature of the extreme point
I found the first order conditions to be:
Equation 1: 4x + 4 - λ = 0
Equation 2: -2y +6 -2λ =0
Equation 3: x+2y-20 =0
![021 =4
ax2
321-2
ay2
²f
ст
=
dxdy dxdy
= 0
Then the matrix becomes:
H
0
1 = [1 −2]
Det (HI) [4-200-2-2]
wwwwww
=
(4-2)(-21)
(4-2)(-2) = 0 and (4)(1 + 2) = 0
The corresponding eigenvalues are:
21=4
2.2=-2
One negative and one positive eigenvalue denotes a saddle point.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F578fbfb4-914e-4f75-b511-b0eb512dadc5%2Fe05f2def-ae48-4d47-8a6e-f0cc2e6e15e5%2F4g3hjjh_processed.jpeg&w=3840&q=75)
Transcribed Image Text:021 =4
ax2
321-2
ay2
²f
ст
=
dxdy dxdy
= 0
Then the matrix becomes:
H
0
1 = [1 −2]
Det (HI) [4-200-2-2]
wwwwww
=
(4-2)(-21)
(4-2)(-2) = 0 and (4)(1 + 2) = 0
The corresponding eigenvalues are:
21=4
2.2=-2
One negative and one positive eigenvalue denotes a saddle point.
Transcribed Image Text:Therefore, it is found that the extreme point for (x,y) is (6,23)
c) Determine the nature of the extreme point
4
To determine the nature of the extreme point, test the second order derivatives
8x2
dxdy
H=
дудх ду²
Therefore, given L(x)= 2x² - y²+4x+6y-λ(x+2y-20)
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