Problem 1 Compute the limit Problem 2 Find global minimizers of lim (x,y) →(0,0) using a Taylor expansion of order 2. on the set X = MA576 - Extra problems Problem 3 Find global minimizers of e²+y(y+1)(1+y+y²/2) x² + y² f(x₁, x₂) = 2x1 +X2, - {(x1, x2) € R² : x² + x² − 1 = 0}. Lagrange multipliers cannot be used. Problem 4 Find global minimizers of f(x₁, x₂) = x₁ - x², on the set X = {(x₁, x₂) € R² : x1 ≤ 2x2 − x², x1 ≥ −x², x1 ≤ 2 − 2x2 − x²}. Lagrange multipliers cannot be used. f(x₁, x2, 3) = exp(x₁ - x₂) + exp(x₂ − x₁) + exp(x²) + x3. ƒ(x1, State if the optima you found is strict or non-strict. Problem 5 The main hypothesis of the FONC condition we studied is that f € C¹ on its domain. We can still use the same optimality condition in the case the function is not C¹ on the whole domain as long as we study separately “problematic” points. For example, the points in which the gradient is undefined. The objective function f(x₁, x2) = √√√x² + x² is continuous in R², that is Cº, but it is not C¹. Find the global minimum of that function. Hint: no need to use FD optimality conditions. Problem 6 Consider minimize xERn = ₂T Ax xBx' X where A and B are symmetric and pd. This problem has infinite solutions, so we choose the solution that satisfies x¹ Bx X 1. This problem can be approached as a regularized least squares problem. The FONC leads to an eigenvalue problem. 1. Why this problem has infinite solutions? 2. Find the stationary points of this problem. 3. Check a second order condition. 4. Solve a particular example in R². 5. How the solution changes if A is psd instead of pd?