(1) Determine the interior, closure, and boundary of each set.
(a) The filled-in ellipse in R2, R = {(x, y) : x2 + 3y2 < 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 - y2} in R3.
(2) Consider the function f that is given by f (x, y) = (x2 - y2)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 - y3 + 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 - y2)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 +z2 = 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 y2 = 9 + xz that are closest to the point (4, 2, 0).
(10) The plane 4x - 3y + 8z = 5 intersects the cone z2 = x2 + y2 in an ellipse. Use Lagrange multipliers to find the highest and lowest points on the ellipse.