Question

6) One of the most powerful inequalities in mathematics is the Cauchy-Bunyakovsky-Schwarz (CBS) inequality. It states: if u, u E R", then lü.히< 111||1| Further, equality is achieved only when ü and v are parallel, i.e. when u = cu for some constant c. Here are several problems that explore the power and scope of CBS. a) Suppose you want to maximize f(x, y, 2) = 8x + 4y +z subject to +y2 + z2 = 1. One way is to use Lagrange multipliers; another is to use CBS. Identify two choices for u and u and use CBS to bound f2 b) This problem is adapted from the famous Putnam competition: Show there is no sequence (an such that for every positive integer m a = a+a+a = k 1 Hint: define a new sequence (bn) = (až) and use CBS. IM8

Answer 1

The expression have to maximi2e fa,9,2) 8x +49+2 C.B.S inegmali ta By Ca +4 + 2 81. 1 8a+49+2 2 b value of 8n +47+2 is 9. So, maximum Comsider the Veetor spae of all infini te b )s.t Sequences With x.2 C.B.S imaqualiy Car a gn thi's Now, Using spaes an for seamene es Cai 2P 2 Za 21 = 4 2. 2.4 9, Con tradiction