Consider min{f (x) : h(x) = 0} satisfying the second order sufficiency conditions. Let p(u) = min{f (x) : h(x) = u}, which is defined for u in an open sphere centered at u = 0. (a) Let f (x) = 1 2 (x2 1 x2 2 x2 3) and h(x) = x1 x2 x3 3. Calculate p(u) and verify that p(0) = , where = 1 is the Lagrange multiplier corresponding to the local minimum x = (1, 1, 1). (b) Consider the augmented Lagrangian function Lc(x, ) = f (x) h(x) c 2 h(x)2. For a given o, let xo argminxRn Lc(x, o). Show that p(uo) = (o ch(xo)), where uo = h(xo)