Math 23bTheoretical Linear Algebra and Multivariable Calculus II PROBLEM SET 9 Problem 1 Use the method of Lagrange multipliers to find the points on the line x y 10 and the ellipse x2 2y 2 1 which are closest Problem 2 Find the points in the closed unit ball B x y z R3 x2 y 2 z 2 1 where the function f x y z x3 y 3 z 3 attains it maximum and minimum Problem 3 Given positive real numbers x1 xn we define their arithmetic and geometric means as follows x1 xn AM n GM n x1 xn Use the Lagrange multipliers to minimize the function x1 xn f x1 xn n on the set S x1 xn Rn x1 xn 1 Use it to prove that the geometric mean is always less than or equal to the arithmetic mean Problem 4 In this problem you are supposed to go over the proofs we have seen in class and write them very carefully on your own making sure that there are no gaps in the arguments Note it is obvious is not an accepted argument Recall that a rectangle R in Rn is a set of type R I1 In where Ik are intervals in R a polygon P is a finite union of rectangles P R1 RN and a partition P of a polygon P is a collection of rectangles P R1 RN such that they are non overlapping Ri Rj for i 6 j and their union is the whole polygon P R1 RN We also define the volume of a rectangle R I1 In to be v R b1 a1 bn an if Ik is an interval with extremes ak and bk a Prove that if R1 RN are arbitrary rectangles in Rn all contained in a bigger rectangle R then there exists a partition P P1 Pk of R consisting of disjoint rectangles such that each of the Ri s is union of some of the Pj s b Prove that if P R1 RN is a polygon in Rn then you can write P as a disjoint union of rectangles Conclude in particular that every polygon admits a partition 1 2 c Prove that if P1 P2 are partitions of the same rectangle R then there exists a refined partition P of R such that every element in P1 and P2 is a union of some of the elements of P d Prove that if R is a rectangle and P R1 RN is any partition of R then v R v R1 v RN e Prove that if P is a polygon and P1 P2 are two partitions of P then X X v R v R R P1 R P2 Hence we can take this as our definition of volume v P of the polygon P f Prove that if P Q are two polygons then v P v Q g Prove that if P R1 RN is a polygon then v P v R1 v RN