MATH 402 Study Guide - Midterm Guide: Convex Function
Document Summary
Presentation: write your answers on the coloured sheets provided. For each xed b > 0, decide whether the extremal x itself provides the optimal path from (0, x(0)) to (b, x(b)). Address all possible types of local minimality: directional, weak, and strong. (expect di erent answers for di erent values of b. ) Hint: the arc y sketched below is an extremal for l obeying y(0) = 0, Note that x(2) may take any real value. x p ws[1,2](cid:26)4x(2)2 +z 2 min. 1 (cid:0)t2 x(t)2 + 2x(t)2(cid:1) dt : x(1) = 1(cid:27) . (b) among all arcs x in p ws[1, 2] obeying both and x(1) = 1. 24x(2)2 +z 2 identify the one that minimizes m [x] def= z 2. Hint: the result from part (a) should help with part (b). Consider the following minimization problem, in which is a constant: x p ws[0,1](cid:26) [x] :=z 1 min.
