2220hw4sol

2220hw4sol - Math 2220 Problem Set 4 Solutions Spring 2010...

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Unformatted text preview: Math 2220 Problem Set 4 Solutions Spring 2010 Section 4.3 : 4. Use Lagrange multipliers to find the critical points of the function f ( x, y ) = xy subject to the constraint 2 x − 3 y = 6. Solution. The critical points are solutions of the system y = λ (2) x = λ ( − 3) 2 x − 3 y = 6 The first two equations give 2 x = − 3 y . Substituting into the third equation, we obtain x = 3 / 2 and y = − 1, so (3 / 2 , − 1) is the only critical point. 8. Use Lagrange multipliers to find the critical points of the function f ( x, y, z ) = x + y + z subject to the constraints y 2 − x 2 = 1 and x + 2 z = 1. Solution. The critical points are solutions of the system 1 = λ ( − 2 x ) + μ (1) 1 = λ (2 y ) + μ (0) 1 = λ (0) + μ (2) y 2 − x 2 = 1 x + 2 z = 1 The third equation gives μ = 1 / 2 and then the first two equations simplify to λx = − 1 / 4 and λy = 1 / 2. These equations imply in particular that x and y are nonzero so we can solve them for λ to get λ = − 1 / 4 x = 1 / 2 y and hence y = − 2 x . Plugging this into the fourth equation gives x = ± radicalbig 1 / 3. Using the fifth equation to obtain the z values, we thus have two critical points ( x, y, z ) = ( radicalbig 1 / 3 , − 2 radicalbig 1 / 3 , (1 − radicalbig 1 / 3) / 2) and ( − radicalbig 1 / 3 , 2 radicalbig 1 / 3 , (1 + radicalbig 1 / 3) / 2) 18. Find the maximum and minimum values of f ( x, y, z ) = x + y − z on the sphere x 2 + y 2 + z 2 = 81. Explain how you know there must be both a maximum and a minimum attained. Solution. The critical points are solutions of the system 1 = λ (2 x ) 1 = λ (2 y ) − 1 = λ (2 z ) x 2 + y 2 + z 2 = 81 1 Math 2220 Problem Set 4 Solutions Spring 2010 The first three equations give that x = y = − z . Substituting into the last equation we get 3 x 2 = 81, i.e., x = 3 √ 3. Thus we have 2 critical points (3 √ 3 , 3 √ 3 , − 3 √ 3) and ( − 3 √ 3 , − 3 √ 3 , 3 √ 3). The values of the function f at these two points are 9 √ 3 and − 9 √ 3....
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2220hw4sol - Math 2220 Problem Set 4 Solutions Spring 2010...

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