Math 226, Fall 2015 Homework 7 Recommended problems (these will not be graded) Section 11.7: # 3, 7, 11, 15, 25, 26, 28,34, 47 Section 11.8: # 3, 4, 8, 15, 18 Required problems (These will be graded) Section 11.7: # 10, 35, 42 Section 11.8: # 7, 16, 21 (1) Determine the interior, closure, and boundary of each set. (a) The lled-in ellipse in R2 , R = {(x, y) : x2 + 3y 2 < 9}. (b) The unit cube in R3 . In other words, the set R = {(x, y, z) : 0 x 1 and 0 y 1 and 0 z 1}. (c) The set {(x, y, z) : z = x2 y 2 } in R3 . (2) Consider the function f that is given by f (x, y) = (x2 y 2 )ex . a) Find all the critical points of f and for each one determine if it corresponds to a local maximum, a local minimum, or a saddle point. If you cannot determine this explain why not. b) Find the global maximum and minimum of f on the triangle whose vertices are (0, 0), (1, 0) and (1, 1). (3) Consider the function f that is given by f (x, y) = x + exy . Find all of its critical points and classify each one as a local maximum, a local minimum or a saddle point. (4) Consider the function f that is given by f (x, y) = x3 y 3 + 6xy. Find all of its critical points and classify each one as a local maximum, a local minimum or a saddle point. (5) Consider the function f that is given by f (x, y) = (x2 y 2 )ex . a) Find all the critical points of f and for each one determine if it corresponds to a local maximum, a local minimum, or a saddle point. If you cannot determine this explain why not. b) Find the global maximum and minimum of f on the triangle whose vertices are (0, 0), (1, 0) and (1, 1). (6) Consider the collection S of points (x, y, z) where x, y and z are all non-negative and x+y +z 2 = 125. Find the maximum and minimum values of the product xyz on S and state where they are attained. Be sure to explain how you know that your answers produce the global maximum and minimum respectively. (7) A cardboard box without a lid is to have a volume of 32 in3 . Find the dimensions that minimize the amount of cardboard used. (8) Find the maximum and minimum values of a rectangular box whose surface area is 1500 cm2 and whose total edge length is 200 cm. (9) Find the points on the surface y 2 = 9 + xz that are closest to the point (4, 2, 0). (10) The plane 4x 3y + 8z = 5 intersects the cone z 2 = x2 + y 2 in an ellipse. Use Lagrange multipliers to nd the highest and lowest points on the ellipse