Consider the following function f: X R f(x, x) = -9x-10x2+0 [-In(100x-x) - In(x) -...

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Consider the following function f: X → R f(x₁, x₂) = -9x₁-10x2+0 [-In(100x₁-x₂) - In(x₁) - ln(x₂) - In(50 - x₁ + x₂)] where is a given parameter and X = {(x1, x₂) x₁ > 0, x₂ > 0, x₁ + x₂ < 100, x1 - x2 < 50}. Suppose you are given the following start points r0 = (8,90)T x⁰ = (1, 40) x⁰ = (15,68)T r0 = (10,20)7 (a) (10pts) Let 0 = 10. What behavior do you observe when you run Newton's method without a line-search to minimize f with these starting points, i.e., by taking a = 1? (b) (15pts) Let 0 = 10. What behavior do you observe when you run Newton's method with a line-search to minimize f with these starting points, i.e., finding an optimal a at each iteration? (c) (10pts) Let 0 = 100. What behavior do you observe when you run Newton's method without a line-search to minimize f with these starting points? (d) (15pts) Let 0 = 100. What behavior do you observe when you run Newton's method with a line-search to minimize f with these starting points? Consider the following function f: X → R f(x₁, x₂) = -9x₁-10x2+0 [-In(100x₁-x₂) - In(x₁) - ln(x₂) - In(50 - x₁ + x₂)] where is a given parameter and X = {(x1, x₂) x₁ > 0, x₂ > 0, x₁ + x₂ < 100, x1 - x2 < 50}. Suppose you are given the following start points r0 = (8,90)T x⁰ = (1, 40) x⁰ = (15,68)T r0 = (10,20)7 (a) (10pts) Let 0 = 10. What behavior do you observe when you run Newton's method without a line-search to minimize f with these starting points, i.e., by taking a = 1? (b) (15pts) Let 0 = 10. What behavior do you observe when you run Newton's method with a line-search to minimize f with these starting points, i.e., finding an optimal a at each iteration? (c) (10pts) Let 0 = 100. What behavior do you observe when you run Newton's method without a line-search to minimize f with these starting points? (d) (15pts) Let 0 = 100. What behavior do you observe when you run Newton's method with a line-search to minimize f with these starting points?