M226q7sol

M226q7sol - g ( x,y,z ) = x 2 + 2 y 2 + 3 z 2-6. The...

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Spring ’11 Math 226 Quiz 7 Solution Name: (15 minutes, no notes/calculators) 1. Find the local maximum and minimum values, and saddle points, of the following function. Identify each point of interest as a minimum, a maximum or a saddle point. f ( x,y ) = x sin( y ) 2. Use Lagrange multipliers to ﬁnd the maximum and minimum values of the function subject to the given constraint. f ( x,y,z ) = xyz, x 2 + 2 y 2 + 3 z 2 = 6 ——————————– Solutions: 1. Using the algorithm from 11.7, ﬁrst we solve the system 5 f = 0 for the critical points: sin( y ) = 0 , x cos( y ) = 0 The solution to the ﬁrst equation is y = (where n is an integer). Since cos( ) = ± 1, we get x = 0 from the second equation. So we have the critical points (0 ,nπ ). We proceed with the second derivative test: f xx = 0 , f yy = - x sin( y ) , f xy = cos( y ) = f yx D ( x,y ) = f xx f yy - f xy f yx = - (cos( y )) 2 D (0 ,nπ ) = - 1 < 0 It follows that we have saddle points at all points (0 ,nπ ). Note f (0 ,nπ ) = 0. 2. We use Lagrange multipliers as in Section 11.8. Set

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Unformatted text preview: g ( x,y,z ) = x 2 + 2 y 2 + 3 z 2-6. The critical points are at the solutions of the system 5 f = 5 g, g ( x,y,z ) = 0 yz = 2 x, xz = 4 y, xy = 6 z, x 2 + 2 y 2 + 3 z 2 = 6 From the rst three equations we get xyz = 2 x 2 , xyz = 4 y 2 , xyz = 6 z 2 xyz = 2 x 2 = 4 y 2 = 6 z 2 x 2 = 2 y 2 = 3 z 2 So from the fourth equation 3 x 2 = 6, or x = 2. Next, y 2 = x 2 2 = 1 = y = 1. And z = q 2 3 . This gives us 8 critical points. The ones with an odd number of negative components yield 1 the minimum, the rest yield the maximum. Maximum: f ( 2 , 1 , s 2 3 ) = f (- 2 ,-1 , s 2 3 ) = f (- 2 , 1 ,-s 2 3 ) = f ( 2 ,-1 ,-s 2 3 ) = 2 3 Minimum: f (- 2 ,-1 ,-s 2 3 ) = f (- 2 , 1 , s 2 3 ) = f ( 2 , 1 ,-s 2 3 ) = f ( 2 , 1 ,-s 2 3 ) =-2 3 2...
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This note was uploaded on 10/15/2011 for the course MATH 39578 taught by Professor Penner during the Spring '09 term at USC.

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M226q7sol - g ( x,y,z ) = x 2 + 2 y 2 + 3 z 2-6. The...

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