Question 11.1. The general quadratic form on R2 is defined as:а bQ(r1; #2) = | x1 x2]ax? + 2bx1.x2 + caс аx2Assume that a > 0 and complete the square to show that Q will always bepositive if both Leading Principal Minors (LPM) of the quadratic coefficientmatrix are larger than 0. (In other words: show that (x1; x2) = (0; 0) is aminimum of Q if the quadratic coefficient matrix is positive definite).1.2. For each of the following quadratic forms, determine formally whetherx = 0 is a minimum, a maximum or a saddlepoint.(a) Q = 10x + 2x% – x1x2.(b) Q = –xỉ – 2.x3 – 2x1x2 along the linear subspace r1 – x2 = 0.

We have to answer the following for the given information.

1.1. The general quadratic form on R² is defined as: a b X1 Q(r1; x2) = | x1 x2 ax² + 2bx1x2 + cx, с d X2 Assume that a > 0 and complete the square to show that Q will always be positive if both Leading Principal Minors (LPM) of the quadratic coefficient matrix are larger than 0. (In other words: show that (x1; x2) = (0; 0) is a %3D minimum of Q if the quadratic coefficient matrix is positive definite). 1.2. For each of the following quadratic forms, determine formally whether x = 0 is a minimum, a maximum or a saddlepoint. %3D (a) Q = 10x² + 2.x3 – x1x2. (b) Q = -xỉ – 2x% – 2x1x2 along the linear subspace x1 – x2 = 0.

1.1. The general quadratic form on R² is defined as: a b X1 Q(r1; x2) = | x1 x2 ax² + 2bx1x2 + cx, с d X2 Assume that a > 0 and complete the square to show that Q will always be positive if both Leading Principal Minors (LPM) of the quadratic coefficient matrix are larger than 0. (In other words: show that (x1; x2) = (0; 0) is a %3D minimum of Q if the quadratic coefficient matrix is positive definite). 1.2. For each of the following quadratic forms, determine formally whether x = 0 is a minimum, a maximum or a saddlepoint. %3D (a) Q = 10x² + 2.x3 – x1x2. (b) Q = -xỉ – 2x% – 2x1x2 along the linear subspace x1 – x2 = 0.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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