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Fall 2007 - Cioaba's Class - Quiz 4

Fall 2007 - Cioaba's Class - Quiz 4 - Name Discussion...

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Name: PID: Discussion Section No: Time: TA’s name: Quiz 4, Math 20C - Lecture D (Fall 2007) Duration: 20 minutes Please close all your notes and books, turn off your phones and put them away. To get full credit you should support your answers. 1. (5 points) Find the absolute maximum and minimum values of f ( x, y ) = x 4 + y 4 4 xy +2 on the set D = { ( x, y ) : 0 x 3 , 0 y 2 } . Solution. First, find the critical points. This is done by solving the system f x ( x, y ) = f y ( x, y ) = 0. Since f x ( x, y ) = 4 x 3 4 y and f y ( x, y ) = 4 y 3 4 x , the previous system is equivalent to the system y = x 3 and x = y 3 . Substituting y from the first equation into second, we get x = y 3 = ( x 3 ) 3 = x 9 which implies 0 = x 9 x = x ( x 8 1). This means that x = 0 or x 8 = 1. The equation x 8 = 1 implies that x = 1 or x = 1. If x = 0, we get y = 0 3 = 0. If x = 1 we get y = ( 1) 3 = 1. If x = 1, we get y = 1 3 = 1. Thus, the solutions of the system f x ( x, y ) = f y ( x, y ) = 0 are (0 , 0) , ( 1 , 1) and (1 , 1). However, only (1 , 1) is a critical point since it is inside domain D . The other two solutions are not critical points. We also have

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Fall 2007 - Cioaba's Class - Quiz 4 - Name Discussion...

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