Extreme Values Subject to Two Constraintsa. Maximum on

Problem 1E Two Independent Variables with One Constraint Extrema on an ellipse Find the points on the ellipse where ƒ(x, y) = xy has its extreme values.

Problem 2E Two Independent Variables with One Constraint Extrema on a circle Find the extreme values of ƒ(x, y) = xy subject to the constraint

Problem 3E Two Independent Variables with One Constraint Maximum on a line Find the maximum value of ƒ(x, y) = 49 - x2 - y2 on the line x + 3y = 10.

Problem 4E Two Independent Variables with One Constraint Extrema on a line Find the local extreme values of ƒ(x, y) = x2 y on the line x + y = 3.

Problem 5E Two Independent Variables with One Constraint Constrained minimum Find the points on the curve xy2 = 54nearest the origin.

Problem 6E Two Independent Variables with One Constraint Constrained minimum Find the points on the curve x2y = 2 nearest the origin.

Problem 7E Two Independent Variables with One Constraint Use the method of Lagrange multipliers to find a. Minimum on a hyperbola The minimum value of x + y subject to the constraints b. Maximum on a line The maximum value of xy, subject to the constraint x + y = 16. Comment on the geometry of each solution.

Problem 8E Two Independent Variables with One Constraint Extrema on a curve Find the points on the curve x2 + xy + y2 = 1 in the xy-plane that are nearest to and farthest from the origin.

Problem 9E Two Independent Variables with One Constraint Minimum surface area with fixed volume Find the dimensions of the closed right circular cylindrical can of smallest surface area whose volume is

Problem 10E Two Independent Variables with One Constraint Cylinder in a sphere Find the radius and height of the open right circular cylinder of largest surface area that can be inscribed in a sphere of radius a. What is the largest surface area?

Problem 11E Two Independent Variables with One Constraint Rectangle of greatest area in an ellipse Use the method of Lagrange multipliers to find the dimensions of the rectangle of greatest area that can be inscribed in the ellipse with sides parallel to the coordinate axes.

Problem 12E Two Independent Variables with One Constraint Rectangle of longest perimeter in an ellipse Find the dimensions of the rectangle of largest perimeter that can be inscribed in the ellipse with sides parallel to the coordinate axes. What is the largest perimeter?

Problem 13E Two Independent Variables with One Constraint Extrema on a circle Find the maximum and minimum values of subject to the constraint

Problem 15E Two Independent Variables with One Constraint Ant on a metal plate The temperature at a point (x, y) on a metal plate is An ant on the plate walks around the circle of radius 5 centered at the origin. What are the highest and lowest temperatures encountered by the ant?

Problem 14E Two Independent Variables with One Constraint Extrema on a circle Find the maximum and minimum values 3x - y + 6 of subject to the constraint

Problem 16E Two Independent Variables with One Constraint Cheapest storage tank Your firm has been asked to design a storage tank for liquid petroleum gas. The customer’s specifications call for a cylindrical tank with hemispherical ends, and the tank is to hold 8000 m3 of gas. The customer also wants to use the smallest amount of material possible in building the tank. What radius and height do you recommend for the cylindrical portion of the tank?

Problem 17E Three Independent Variables with One Constraint Minimum distance to a point Find the point on the plane x + 2y + 3z = 13 closest to the point (1, 1, 1).

Problem 18E Three Independent Variables with One Constraint Maximum distance to a point Find the point on the sphere farthest from the point (1, -1, 1).

Problem 19E Three Independent Variables with One Constraint Minimum distance to the origin Find the minimum distance from the surface to the origin.

Problem 20E Three Independent Variables with One Constraint Minimum distance to the origin Find the point on the surface z = xy + 1 nearest the origin.

Problem 21E Three Independent Variables with One Constraint Minimum distance to the origin Find the points on the surface z2 = xy + 4closest to the origin.

Problem 22E Three Independent Variables with One Constraint Minimum distance to the origin Find the point(s) on the surface xyz = 1 closest to the origin.

Problem 23E Three Independent Variables with One Constraint Extrema on a sphere Find the maximum and minimum values of ƒ(x, y, z) = x - 2y + 5z on the sphere

Problem 24E Three Independent Variables with One Constraint Extrema on a sphere Find the points on the sphere has its maximum and minimum values.

Problem 25E Three Independent Variables with One Constraint Minimizing a sum of squares Find three real numbers whose sum is 9 and the sum of whose squares is as small as possible.

Problem 26E Three Independent Variables with One Constraint Maximizing a product Find the largest product the positive numbers x, y, and z can have if x + y + z2 = 16.

Problem 27E Three Independent Variables with One Constraint Rectangular box of largest volume in a sphere Find the dimensions of the closed rectangular box with maximum volume that can be inscribed in the unit sphere.

Problem 28E Three Independent Variables with One Constraint Box with vertex on a plane Find the volume of the largest closed rectangular box in the first octant having three faces in the coordinate planes and a vertex on the plane x/a + y/b + z/c = 1,where a > 0, b > 0, and c > 0

Problem 29E Three Independent Variables with One Constraint Hottest point on a space probe A space probe in the shape of the ellipsoid enters Earth’s atmosphere and its surface begins to heat. After 1 hour, the temperature at the point (x, y, z) on the probe’s surface is Find the hottest point on the probe’s surface.

Problem 31E Cobb-Bonging production function During the 1920s, Charles Cobb and Paul Douglas modeled total production output P (of a firm, industry, or entire economy) as a function of labor hours involved x and capital invested y (which includes the monetary worth of all buildings and equipment). The Cobb-Douglas production function is given by where k and ? are constants representative of a particular firm or economy. a. Show that a doubling of both labor and capital results in a doubling of production P. b. Suppose a particular firm has the production function for k = 120 and ? = 3/4. Assume that each unit of labor costs $250 and each unit of capital costs $400, and that the total expenses for all costs cannot exceed $100,000. Find the maximum production level for the firm.

Problem 30E Three Independent Variables with One Constraint Extreme temperatures on a sphere Suppose that the Celsius temperature at the point (x, y, z) on the sphere x2 + y2 + z2 = 1 is T = 400xyz2 Locate the highest and lowest temperatures on the sphere.

Problem 32E (Continuation of Exercise 31.) If the cost of a unit of labor is and the cost of a unit of capital is and if the firm can spend only B dollars as its total budget, then production P is constrained by . Show that the maximum production level subject to the constraint occurs at the point Reference: Cobb-Bonging production function During the 1920s, Charles Cobb and Paul Douglas modeled total production output P (of a firm, industry, or entire economy) as a function of labor hours involved x and capital invested y (which includes the monetary worth of all buildings and equipment). The Cobb-Douglas production function is given by where k and ? are constants representative of a particular firm or economy. a. Show that a doubling of both labor and capital results in a doubling of production P. b. Suppose a particular firm has the production function for k = 120 and ? = 3/4. Assume that each unit of labor costs $250 and each unit of capital costs $400, and that the total expenses for all costs cannot exceed $100,000. Find the maximum production level for the firm.

Problem 33E Three Independent Variables with One Constraint Maximizing a utility function: an example from economics In economics, the usefulness or utility of amounts x and y of two capital goods G1 and G2 is sometimes measured by a function U(x, y). For example, G1 and G2 might be two chemicals a pharmaceutical company needs to have on hand and U(x, y) the gain from manufacturing a product whose synthesis requires different amounts of the chemicals depending on the process used. If G1 costs a dollars per kilogram, G2 costs b dollars per kilogram, and the total amount allocated for the purchase of G1 and G2 together is c dollars, then the company’s managers want to maximize U(x, y) given that ax + by = c .Thus, they need to solve a typical Lagrange multiplier problem. Suppose that U(x, y) = xy + 2x and that the equation ax + by = c simplifies to 2x + y = 30. Find the maximum value of U and the corresponding values of x and y subject to this latter constraint.

Problem 34E Blood types Human blood types are classified by three gene 0 forms A, B, and 0. Blood types AA, BB, and 00 are homozygous, and blood types AB, AO, and BO are heterozygous. If p, q, and r represent the proportions of the three gene forms to the population, respectively, then the Hardy-Weinberg Law asserts that the proportion Q of heterozygous persons in any specific population is modeled by Q(p, q, r) = 2(pq + pr + qr), subject to p + q + r = 1. Find the maximum value of Q.

Problem 35E Length of a beam In Section 4.6, Exercise 39, we posed a problem of finding the length L of the shortest beam that can reach over a wall of height h to a tall building located k units from the wall. Use Lagrange multipliers to show that

Problem 36E Three Independent Variables with One Constraint Locating a radio telescope You are in charge of erecting a radio telescope on a newly discovered planet. To minimize interference, you want to place it where the magnetic field of the planet is weakest. The planet is spherical, with a radius of 6 units. Based on a coordinate system whose origin is at the center of the planet, the strength of the magnetic field is given by M(x, y, z) = 6x - y2 + xz + 60.Where should you locate the radio telescope?

Problem 37E Extreme Values Subject to Two Constraints Maximize the function subject to the constraints 2x - y = 0 and y + z = 0.

Problem 38E Extreme Values Subject to Two Constraints Minimize the function subject to the constraints x + 2y + 3z = 6and x + 3y + 9z = 9.

Problem 39E Extreme Values Subject to Two Constraints Minimum distance to the origin Find the point closest to the origin on the line of intersection of the planes y + 2z = 12 and x + y = 6.

Problem 40E Extreme Values Subject to Two Constraints Maximum value on line of intersection Find the maximum value that can have on the line of intersection of the planes 2x - y = 0 and y + z = 0.

Problem 41E Extreme Values Subject to Two Constraints Extrema on a curve of intersection Find the extreme values of on the intersection of the plane with the sphere

Problem 42E Extreme Values Subject to Two Constraints a. Maximum on line of intersection Find the maximum value of on the line of intersection of the two planes x + y + z = 40 and x + y - z = 0. b. Give a geometric argument to support your claim that you have found a maximum, and not a minimum, value of w.

Problem 43E Extreme Values Subject to Two Constraints Extrema on a circle of intersection Find the extreme values of the function on the circle in which the plane y - x = 0 intersects the sphere

Problem 44E Extreme Values Subject to Two Constraints Minimum distance to the origin Find the point closest to the origin on the curve of intersection of the plane 2y + 4z = 5 and the cone

Problem 45E Theory and Examples The condition , it does not in itself guarantee that one exists. As a case in point, try using the method of Lagrange multipliers to find a maximum value of ƒ(x, y) = x + y subject to the constraint that xy = 16 . The method will identify the two points (4, 4) and ( -4, -4) as candidates for the location of extreme values. Yet the sum (x + y) has no maximum value on the hyperbola xy = 16 The farther you go from the origin on this hyperbola in the first quadrant, the larger the sum ƒ(x, y) = x + y becomes.

A least squares plane The plane \(z=A x+B y+C\) is to be "fitted" to the following points \(\left(x_{k}, y_{k}, z_{k}\right)\): , Find the values of , and that minimize \(\sum_{k=1}^{4}\left(A x_{k}+B y_{k}+C-z_{k}\right)^{2}\) the sum of the squares of the deviations. Equation Transcription: Text Transcription: z=A x+B y+C \left(x_{k}, y_{k}, z_{k}\right) \sum_{k=1}^{4}\left(A x_{k}+B y_{k}+C-z_{k}\right)^{2}

Problem 47E Theory and Examples a. Maximum on a sphere Show that the maximum value of a2b2c2 on a sphere of radius r centered at the origin of a Cartesian abc-coordinate system is b. Geometric and arithmetic means Using part (a), show that for nonnegative numbers a, b, and c, that is, the geometric mean of three nonnegative numbers isless than or equal to their arithmetic mean.

Problem 48E Theory and Examples Sum of products Let a1, a2 , … , an be n positive numbers. Find the maximum of

Problem 49E COMPUTER EXPLORATIONS In Exercise, use a CAS to perform the following steps implementing the method of Lagrange multipliers for finding constrained extrema: a. Form the function where ƒ is the function to optimize subject to the constraints g1 = 0 and g2 = 0. b. Determine all the first partial derivatives of h, including the partials with respect to and set them equal to 0. c. Solve the system of equations found in part (b) for all the unknowns, including d. Evaluate ƒ at each of the solution points found in part (c) and select the extreme value subject to the constraints asked for in the exercise. Minimize ƒ(x, y, z) = xy + yz subject to the constraints x2 + y2 - 2 = 0 and x2 + z2 - 2 = 0.

Problem 50E COMPUTER EXPLORATIONS In Exercise, use a CAS to perform the following steps implementing the method of Lagrange multipliers for finding constrained extrema: a. Form the function where ƒ is the function to optimize subject to the constraints g1 = 0 and g2 = 0. b. Determine all the first partial derivatives of h, including the partials with respect to and set them equal to 0. c. Solve the system of equations found in part (b) for all the unknowns, including d. Evaluate ƒ at each of the solution points found in part (c) and select the extreme value subject to the constraints asked for in the exercise. Minimize subject to the constraints

Problem 51E COMPUTER EXPLORATIONS In Exercise, use a CAS to perform the following steps implementing the method of Lagrange multipliers for finding constrained extrema: a. Form the function where ƒ is the function to optimize subject to the constraints g1 = 0 and g2 = 0. b. Determine all the first partial derivatives of h, including the partials with respect to and set them equal to 0. c. Solve the system of equations found in part (b) for all the unknowns, including d. Evaluate ƒ at each of the solution points found in part (c) and select the extreme value subject to the constraints asked for in the exercise. Maximize subject to the constraints

Problem 52E COMPUTER EXPLORATIONS In Exercise, use a CAS to perform the following steps implementing the method of Lagrange multipliers for finding constrained extrema: a. Form the function where ƒ is the function to optimize subject to the constraints g1 = 0 and g2 = 0. b. Determine all the first partial derivatives of h, including the partials with respect to and set them equal to 0. c. Solve the system of equations found in part (b) for all the unknowns, including d. Evaluate ƒ at each of the solution points found in part (c) and select the extreme value subject to the constraints asked for in the exercise. Minimize subject to the constraints

Problem 53E COMPUTER EXPLORATIONS In Exercise, use a CAS to perform the following steps implementing the method of Lagrange multipliers for finding constrained extrema: a. Form the function where ƒ is the function to optimize subject to the constraints g1 = 0 and g2 = 0. b. Determine all the first partial derivatives of h, including the partials with respect to and set them equal to 0. c. Solve the system of equations found in part (b) for all the unknowns, including d. Evaluate ƒ at each of the solution points found in part (c) and select the extreme value subject to the constraints asked for in the exercise. Minimize subject to the constraints 2x - y + z - w - 1 = 0 and x + y - z + w - 1 = 0.

Problem 54E COMPUTER EXPLORATIONS In Exercise, use a CAS to perform the following steps implementing the method of Lagrange multipliers for finding constrained extrema: a. Form the function where ƒ is the function to optimize subject to the constraints g1 = 0 and g2 = 0. b. Determine all the first partial derivatives of h, including the partials with respect to and set them equal to 0. c. Solve the system of equations found in part (b) for all the unknowns, including d. Evaluate ƒ at each of the solution points found in part (c) and select the extreme value subject to the constraints asked for in the exercise. Determine the distance from the line y = x + 1 to the parabola y2 = x. (Hint: Let (x, y) be a point on the line and (w, z) a point on the parabola. You want to minimize (x – w)2 + (y – z)2.)