Solve the following geometric problems by Lagranges method. (a) Find the shortest distance from the point (a1, a2, a3) in R3 to the plane whose equation is given by b1x1 + b2x2 + b3x3 + b0 = 0, where (b1, b2, b3) = (0, 0, 0). (b) Find the point on the line of intersection of the two planes a1x1 + a2x2 + a3x3 = 0 and b1x1 + b2x2 + b3x3 + b0 = 0 that is nearest to the origin. (c) Show that the volume of the largest rectangular parallelepiped that can be inscribed in the ellipsoid x2 a2 + y2 b2 + z2 c2 = 1 is 8abc/3 3.

MARKET EFFICIENCY ch 3,4 7(skim) (“market efficiency” posted under extra readings tab on Bb) Prelim: measuring efficiency - Recall that we evaluate actions “at the margin” o The issue is not whether an action is “good” ro “bad” but whether a little more action will generate more benefits than costs - Consider a patent, an exclusive right to sell a product for 20 yrs o The benefit of a patnt is that it provides firms and induviduals with the incentive to develop new and better products o The cost of a patent is that it reduces competition, and thus allows patent holders to charge higher prices - Would it be efficient to have longer patent protection o If extending the duration of