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
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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.
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.
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)
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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