6. [36 points] Let f: R² → R be defined by f(x, y) = (x² − 1)² + (x² + y − 1)² (a) [8 points] What is the gradient Vf at

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6 36 Points Let F R R Be Defined By F X Y X 1 X Y 1 A 8 Points What Is The Gradient Vf At 1 (68.8 KiB) Viewed 37 times

6. [36 points] Let f: R² → R be defined by f(x, y) = (x² − 1)² + (x² + y − 1)² (a) [8 points] What is the gradient Vf at (x, y)? (b) [12 points] What is the Hessian matrix V²ƒ at (x, y)? (c) [6 points] Find all critical points and determine them if each is local minimum/maximum point or saddle point. (d) [10 points] With initial xo = (0.1, 0.5), to apply the Newton's method (with a constant step size a = 1), write the formula for ₁ and compute the value of ₁. Is the search direction here descent? If not, to apply Levenberg-Marquardt modification, the Hessian matrix H will be added by a scalar matrix µI. Determine a value of such that the modified search direction in this way is a descent direction.