Engineering Calculus Notes 402

Engineering Calculus Notes 402 - Q(a Q x,y = 17 x 2 12 xy 8...

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390 CHAPTER 3. REAL-VALUED FUNCTIONS: DIFFERENTIATION The second partials are f xx = 4 f xy = 2 f xz = 2 f yy = 4 f yz = 2 f zz = 2 so Δ 3 = det 4 2 2 2 4 2 2 2 2 = 4(8 4) 2(4 4) + 2(4 8) = 16 0 8 = 8 > 0 Δ 2 = (4)(4) (2)(2) = 12 Δ 1 = 4 > 0 so the Hessian is positive defnite , and f has a local minimum at (3 , 1 , 2). Exercises for § 3.9 Practice problems: 1. For each quadratic form Q below, (i) write down its matrix representative [ Q ]; (ii) ±nd all eigenvalues of [ Q ]; (iii) ±nd corresponding unit eigenvectors; (iv) write down the weighted squares representative of
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Unformatted text preview: Q . (a) Q ( x,y ) = 17 x 2 + 12 xy + 8 y 2 (b) Q ( x,y ) = 11 x 2 + 6 xy + 19 y 2 (c) Q ( x,y ) = 3 x 2 + 4 xy (d) Q ( x,y ) = 19 x 2 + 24 xy + y 2 (e) Q ( x,y,z ) = 6 x 2 − 4 xy + 6 y 2 + z 2 (f) Q ( x,y,z ) = 2 x 2 + y 2 + z 2 + 2 xy − 2 xz (g) Q ( x,y,z ) = 5 x 2 + 3 y 2 + 3 z 2 + 2 xy − 2 xz − 2 yz 2. For each function below, ±nd all critical points and classify each as local minimum, local maximum, or neither....
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