P14-11-8

P14-11-8 - x = 2 z , we have x = 2 X 1 6 ~ = 2 6 . At this...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
Department of Mathematics University of Southern California March 4, 2011 Class Problem; Section 11-8#14 Find the minimum and the maximum of the function f ( x, y, z )=3 x y 3 z, (1) subject to two constraints g ( x, y, z )= x + y z =0 , (2) and h ( x, y, z )= x 2 +2 z 2 =1 . (3) Let us f rst obtain the gradients, f = < 3 , 1 , 3 > g = < 1 , 1 , 1 > h = < 2 x, 0 , 4 z> At extremum values, f = λ g + μ h, i.e., < 3 , 1 , 3 > = λ < 1 , 1 , 1 > + μ < 2 x, 0 , 4 z>. Equivalently, this represents the three equations, 3= λ +2 μ x (4) 1= λ (5) 3= λ +4 μ z (6) From equation (5), we determine λ = 1. Using this information in equations (4) and (6) yields, 3= 1+2 μ x, 3=1+ 4 μ z, which simplify to 4=2 μ x, (7) 4=4 μ z. (8) Now, if we divide equation (7) by (8), provided μ W =0,we f nd 1= x 2 z , i.e., x = 2 z. The use of this result in the second constraint (3) leads to h ( x, y, z )= x 2 +2 z 2 =4 z 2 +2 z 2 =6 z 2 =1 .
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This gives us the possible values of z as z = ± 1 6 , and with
Background image of page 2
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: x = 2 z , we have x = 2 X 1 6 ~ = 2 6 . At this point, we can con f rm the value of . From equation (7), = 2 x = 6 W = 0 . With the f rst constraint (2), the value of y can be found as y = x + z = X 2 6 ~ 1 6 = 2 6 1 6 = 3 6 . Now with ( x, y, z ) determined as ( x, y, z ) = X 2 6 , 3 6 , 1 6 ~ , at the extrema, f = 3 x y 3 z = 3 X 2 6 ~ X 3 6 ~ 3 X 1 6 ~ = 6 3 3 6 = 12 6 = 2 6 From here, we may infer that the maximum for f ( x, y, z ), under the constraints (2) and (3) is 2 6 and the minimum is 2 6. 2...
View Full Document

This note was uploaded on 10/15/2011 for the course MATH 39578 taught by Professor Penner during the Spring '09 term at USC.

Page1 / 2

P14-11-8 - x = 2 z , we have x = 2 X 1 6 ~ = 2 6 . At this...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online