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Engineering Calculus Notes 400

Engineering Calculus Notes 400 - 10 − = − 3 16 2...

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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 definite; Δ 2 = ( 3)( 3) (1)(1) = 8 > 0 which is still consistent with being negative definite, and finally Δ 1 = 3 < 0; we see that Q satisfied the conditions of Corollary 3.9.7 , and so it is negative definite . Note that the characteristic polynomial of [ Q ] is det 3 λ 1 4 1 3 λ 2 4
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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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