2220hw4 - Math 2220 Problem Set 4 Spring 2010 Section 4.3 :...

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Unformatted text preview: Math 2220 Problem Set 4 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. 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. 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. 20. Find the dimensions of the rectangular box of maximum volume subject to the con- straint that the sum of the length plus the girth of the box is at most 108 inches. (The girth is the total circumference of a rectangular cross section perpendicular to the length direction. Thus the girth is twice the width plus twice the height of the box.) 22. Suppose one is building a silo to hold 900 cubic feet of grain. The silo is to be cylindrical in shape with a hemispherical roof and a flat floor. Suppose it costs five timescylindrical in shape with a hemispherical roof and a flat floor....
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