Engineering Calculus Notes 400

Engineering - 10 − = − 3 16 2 48 4 which has no obvious factorization(in fact it has no integer zeroes Thus we can determine that the

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388 CHAPTER 3. REAL-VALUED FUNCTIONS: DIFFERENTIATION representative [ Q ] = 3 1 4 1 3 2 4 2 10 . The determinant of this matrix is Δ 3 = det[ Q ] = ( 3)[( 3)( 10) (2)(2)] (1)[(1)( 10) (4)(2)] + (4)[(1)(2) (4)( 3)] = ( 3)[26] [ 18] + (4)[2 + 12] = 78 + 18 + 56 = 4 < 0 so the form is not positive deFnite; Δ 2 = ( 3)( 3) (1)(1) = 8 > 0 which is still consistent with being negative deFnite, and Fnally Δ 1 = 3 < 0; we see that Q satisFed the conditions of Corollary 3.9.7 , and so it is negative defnite . Note that the characteristic polynomial of [ Q ] is det 3 λ 1 4 1 3 λ 2 4 2
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Unformatted text preview: 10 − λ = − ( λ 3 + 16 λ 2 + 48 λ + 4) which has no obvious factorization (in fact, it has no integer zeroes). Thus we can determine that the form is negative deFnite far more easily than we can calculate its weighted squares expression. Combining the analysis in Proposition 3.9.6 and Corollary 3.9.7 with Proposition 3.8.3 and Lemma 3.8.4 , we can get the three-variable analogue of the Second Derivative Test which we obtained for two variables in Theorem 3.8.6 :...
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This note was uploaded on 10/20/2011 for the course MAC 2311 taught by Professor All during the Spring '08 term at University of Florida.

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