K QX k T Q Q X k 6 iii For the function fX x 1 2 4x 1 x 2 x 2 2 Use the optimum

K qx k t q q x k 6 iii for the function fx x 1 2 4x 1

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K (QX k ) T Q Q X k 6 iii) For the function f(X) = x 1 2 + 4x 1 x 2 + x 2 2 Use the optimum step length derived in a)ii) to calculate X 1 given that: X 0 = [ 2 2 ] T 5 c) i) Given the quadratic function f(X) = C T X + 1 2 X T Q X Show that the optimum solution can be sought in a single iteration by using the Newton- Raphson method (refer to Appendix) such that the optimum solution X* = -Q -1 C. 5 ii) A system has two state equations such that: w ̇ = x 1 + 5x 2 + 1 v ̈ = x 1 + 2x 2 By using the result of 4c i) calculate the values of x 1 and x 2 that minimise the system property f(X) given by: f(X) = w ̇ v ̈ 12x 1 x 2 Prove that the solution is a minimum by evaluating if Q is positive definite, negative definite or indefinite. 6 Total 25
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