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Fall 2007 - Rabin's Class - Exam 2 (Version A)

Fall 2007 - Rabin's Class - Exam 2 (Version A) - Math 20C...

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Math 20C – Prof. Rabin – Second Midterm Exam, Version A – November 28, 2007 Write your name and your section number or time at the top of this exam. You may use one page of notes, but no calculator or any other assistance. If any question is not clear, ask the instructor for clarification. Show all work necessary to justify your answers. Please box your answers and cross out any “false starts”. Each problem is worth 25 points. Good luck! (1) Find any local maximum or minimum points of the function f ( x, y ) = x 3 + y 2 - 6 xy . The first step is to find the critical points, the points where ∂f ∂x and ∂f ∂y are both 0. Computing these first partial derivatives gives: f x = 3 x 2 - 6 y and f y = 2 y - 6 x. We solve for x by solving f y = 2 y - 6 x = 0 for y , which is y = 3 x , and substituting in this in for f x = 3 x 2 - 6 y = 0 producing 3 x 2 - 6 y = 0 3 x 2 - 6(3 x ) = 0 3( x - 6) x = 0 which has solutions x = 0 and x = 6. Using, these values for x in either equation gives (0 , 0) and (6 , 18) as the critical points of the curve. One must now determine whether these critical points are saddle points or local extrema by examining the sign of D = f xx f yy - ( f xy ) 2 . First we compute the necessary second partial derivatives: f xx = ∂x [3 x 2 - 6 y ] = 6 x, f yy = ∂y [2 y - 6 x ] = 2 , f xy = ∂y [3 x 2 - 6 y ] = - 6 .
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