Question

1. Consider the problem minimize f (x1, x2) = x} + 2x3 – 21 – 4x2 + 2. (a) (4 points) Find all of the points (21, x2)T that satisfy the first-order necessary condition (FONC). (b) (4 points) For each of the points in the above question, identify whether it a local minimizer, local maximizer, or saddle point. (c) (2 points) Is there a global minimizer?

Answer 1

Anss minimize la) of Ix f (x,x2) = x3 + 2x3 _x, - 3x,- 4X2+2 3x21 x = + F = 0 =) af = 6x₂ - 4 af ax, 2 d22 26 = 07 X2 = ± 5 3 3 3 X2 - critical points the te points are f - 68, 41 (b) da, 2 = 0 =s 32f Эх,2х, 2²f 2x2 =t = 12x2 for litir 2.13 6X-3 r = &=0 2. t = 12x8 = 4X56 = 456 rt 70 r = 253 70 -8 It') f hors local minima at Foto 3) r = -253 s-0 456 t =

is local maxima point 2 IS saddle point for rt_s2 20 ( - 1/3 1 5 3 3 1-33) r = 253 s=0 t = – 456 It! rt-82 Lo ( 73 - ) is saddle point for l-ti- r = – 253 8 = 0 456 + yo 2. Mts rzo local minimazar (-1/2 1.5 (C) Because there is that is also only one global mini anizas a that is
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