# self quiz 6 answers - MATH 210 Self-quiz 6 1 Find and...

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MATH 210 Self-quiz 6 March 01, 2009 1. Find and classify the critical points of the function f ( x, y ) = x 3 - 3 xy + y 3 . Solution: We need to ﬁnd the ﬁrst partial derivatives and set them equal to 0. We have: f x ( x, y ) = 3 x 2 - 3 y f y ( x, y ) = 3 y 2 - 3 x If we set the above two functions equal to 0, then we get: x 2 - y = 0 y 2 - x = 0 This means that; x 2 = y y 2 = x By subsituting the expression for y from the ﬁrst equation, into the second we get: x 4 - x = 0. This can be factorized as: x ( x - 1)( x 2 + x + 1) = 0 that has roots 0 , 1. If x = 0 then y = 0 (by the ﬁrst equation). Also if x = 1, then y = 1. We, thus, get two critical points (0 , 0) and (1 , 1). We will apply the 2nd derivative test for each one of them. To this end we need to compute the 2nd order partial derivatives. We have: f xx ( x, y ) = 6 x - 3 f xy ( x, y ) = - 3 f yy ( x, y ) = 6 y - 3 We begin our computation with (0 , 0). At that point, we get that:

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D = ± ± ± ± - 3 - 3 - 3 - 3 ± ± ± ± = 0 The 2nd derivative test is inconclusive for this point. The dominant term
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## This note was uploaded on 10/27/2009 for the course MATH 210 taught by Professor Slodowski during the Spring '08 term at Ill. Chicago.

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self quiz 6 answers - MATH 210 Self-quiz 6 1 Find and...

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