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Problem 1 The objective of this exercise is to classify the stationary points locally and globally and observe the effect of the restrictions. Later in the course we will use a more systematic way by the Lagrangian and Karush-Kuhn-Tucker conditions. a) b) where a € R. c) minimize 4x¹2³ — 4x² +1 s.t. z € [-1,1]. minimize (za)² +1 s.t. 2 € [-1,1], minimize ||rc|² +1 s.t. z = [0, 1]², where z € R², c = (2,1/2) and [0, 1]² = {(₁, ₂) € R² : 0 ≤ ₁ ≤ 1, 0 ≤ ₂ ≤ 1}. d) minimize (x - 2)² + y²-y s.t. x+y≤ 4 r≥ 0, y ≥ 0. Hint: Plotting the contour lines of the objetive function and the feasible region in each case may help to solve the problem.

Fig: 1