A biological rule of thumb states that as the area A of anisland increases tenfold, the

For Exercises 16, find the critical points of the given function and classify them as local maxima, local minima, saddle points, or none of these. f(x, y) = x2 + 2xy y2 4x 8y + 9

For Exercises 16, find the critical points of the given function and classify them as local maxima, local minima, saddle points, or none of these. f(x, y)=2xy2 x2 2y2 + 1

For Exercises 16, find the critical points of the given function and classify them as local maxima, local minima, saddle points, or none of these. f(x, y) = x3 + y2 3x2 + 10y +6

For Exercises 16, find the critical points of the given function and classify them as local maxima, local minima, saddle points, or none of these. f(x, y) = x2y + 2y2 2xy + 6

For Exercises 16, find the critical points of the given function and classify them as local maxima, local minima, saddle points, or none of these. f(x, y) = 80 xy + 10x + 10xy + 20y

For Exercises 16, find the critical points of the given function and classify them as local maxima, local minima, saddle points, or none of these. f(x, y) = sin x + sin y + sin(x + y), 0 <x<, 0 <y<.

For Exercises 710, find the local maxima, local minima, and saddle points of the function. Decide if the local maxima or minima are global maxima or minima. Explain. f(x, y) = 10 + 12x + 6y 3x2 y2

For Exercises 710, find the local maxima, local minima, and saddle points of the function. Decide if the local maxima or minima are global maxima or minima. Explain. f(x, y) = x2 + y3 3xy

For Exercises 710, find the local maxima, local minima, and saddle points of the function. Decide if the local maxima or minima are global maxima or minima. Explain. f(x, y) = x + y + 1 x + 4 y

For Exercises 710, find the local maxima, local minima, and saddle points of the function. Decide if the local maxima or minima are global maxima or minima. Explain. f(x, y) = xy + ln x + y2 10, x> 0

For Exercises 1122, use Lagrange multipliers to find the maximum and minimum values of f subject to the constraint. f(x, y)=3x 4y, x2 + y2 = 5

For Exercises 1122, use Lagrange multipliers to find the maximum and minimum values of f subject to the constraint. f(x, y) = x2 + y2, x4 + y4 = 2

For Exercises 1122, use Lagrange multipliers to find the maximum and minimum values of f subject to the constraint. f(x, y) = x2 + y2, 4x 2y = 15

For Exercises 1122, use Lagrange multipliers to find the maximum and minimum values of f subject to the constraint. f(x, y) = x2 xy + y2, x2 y2 = 1

For Exercises 1122, use Lagrange multipliers to find the maximum and minimum values of f subject to the constraint. f(x, y) = x2 + 2y2, 3x + 5y = 200

For Exercises 1122, use Lagrange multipliers to find the maximum and minimum values of f subject to the constraint. f(x, y) = xy, 4x2 + y2 = 8

For Exercises 1122, use Lagrange multipliers to find the maximum and minimum values of f subject to the constraint. f(x, y) = 3x2 2y2 + 20xy, x + y = 100

For Exercises 1122, use Lagrange multipliers to find the maximum and minimum values of f subject to the constraint. f(x, y, z) = x2 2y + 2z2, x2 + y2 + z2 = 1

For Exercises 1122, use Lagrange multipliers to find the maximum and minimum values of f subject to the constraint. f(x, y, z)=2x + y + 4z, x2 + y + z2 = 16

For Exercises 1122, use Lagrange multipliers to find the maximum and minimum values of f subject to the constraint. z = 4x2 xy + 4y2, x2 + y2 2

For Exercises 1122, use Lagrange multipliers to find the maximum and minimum values of f subject to the constraint. f(x, y) = x2 y2, x2 y

For Exercises 1122, use Lagrange multipliers to find the maximum and minimum values of f subject to the constraint. f(x, y) = x3 y2, x2 + y2 1

In Exercises 2326, does f(x, y) = x2y2 have a maximum, a minimum, neither, or both when subject to the constraint? x = 10

In Exercises 2326, does f(x, y) = x2y2 have a maximum, a minimum, neither, or both when subject to the constraint? y = 10

In Exercises 2326, does f(x, y) = x2y2 have a maximum, a minimum, neither, or both when subject to the constraint? x2 + y2 = 10

In Exercises 2326, does f(x, y) = x2y2 have a maximum, a minimum, neither, or both when subject to the constraint? xy = 10

Maximize 0.3 ln x + 0.7 ln y on 2x + 3y = 6.

(a) Write an expression for the distance between the points (3, 4) and (x, y). (b) Minimize this distance if (x, y)lies on the unit circle centered at the origin. At what point does the minimum occur? (c) What is the maximum distance? At what point does it occur?

If the right side of the constraint in Exercise 12 is changed by the small amount c, by approximately how much do the maximum and minimum values change?

Compute the regression line for the points (1, 2), (0, 1), (1, 1) using least squares.

At the point (1, 3), suppose fx = fy = 0 and fxx < 0, fyy < 0, fxy = 0. Draw a possible contour diagram.

For f(x, y) = A (x2 + Bx + y2 + Cy), what values of A, B, and C give f a local maximum value of 15 at the point (2, 1)?

A biological rule of thumb states that as the area A of an island increases tenfold, the number of animal species, N, living on it doubles. The table contains data for islands in the West Indies. Assume that N is a power function of A. (a) Use the biological rule of thumb to find (i) N as a function of A (ii) ln N as a function of ln A (b) Using the data given, tabulate ln N against ln A and find the line of best fit. Does your answer agree with the biological rule of thumb?

A firm manufactures a commodity at two different factories. The total cost of manufacturing depends on the quantities, q1 and q2, supplied by each factory, and is expressed by the joint cost function, C = f(q1, q2)=2q2 1 + q1q2 + q2 2 + 500. The companys objective is to produce 200 units, while minimizing production costs. How many units should be supplied by each factory?

(a) Let f(x, y) = x2 + 2y2. Find the minimum value m(c) of f on the line x + y = c as a function of c. (b) Give the value of the Lagrange multiplier at this minimum. (c) What is the relation between your answers in parts (a) and (b)?

The maximum value of f(x, y) subject to the constraint g(x, y) = 240 is 6300. The method of Lagrange multipliers gives = 20. Find an approximate value for the maximum of f(x, y) subject to the constraint g(x, y) = 242.

An industry manufactures a product from two raw materials. The quantity produced, Q, can be given by the Cobb-Douglas function: Q = cxayb , where x and y are quantities of each of the two raw materials used and a, b, and c are positive constants. The first raw material costs $P1 per unit and the second costs $P2 per unit. Find the maximum production possible if no more than $K can be spent on raw materials.

The quantity, Q, of a product manufactured by a company is given by Q = aK0.6 L0.4 , where a is a positive constant, K is the quantity of capital and L is the quantity of labor used. Capital costs are $20 per unit, labor costs are $10 per unit, and the company wants costs for capital and labor combined to be no higher than $150. Suppose you are asked to consult for the company, and learn that 5 units each of capital and labor are being used. (a) What do you advise? Should the company use more or less labor? More or less capital? If so, by how much? (b) Write a one-sentence summary that could be used to sell your advice to the board of directors.

Figure 15.40 shows contours labeled with values of f(x, y) and a constraint g(x, y) = c. Mark the approximate points at which: (a) grad f = grad g (b) f has a maximum (c) f has a maximum on the constraint g = c.

A mountain climber at the summit of a mountain wants to descend to a lower altitude as fast as possible. The altitude of the mountain is given approximately by h(x, y) = 3000 1 10,000 (5x2 + 4xy + 2y2 ) meters, where x, y are horizontal coordinates on the earth (in meters), with the mountain summit located above the origin. In thirty minutes, the climber can reach any point (x, y) on a circle of radius 1000 m. In which direction should she travel in order to descend as far as possible?

Let f(x, y, z) = (x a)2 + (y b)2 + (z c)2. Minimize f subject to Ax + By + Cz + D = 0. What is the geometric meaning of your solution?

A company sells two products which are partial substitutes for each other, such as coffee and tea. If the price of one product rises, then the demand for the other product rises. The quantities demanded, q1 and q2, are given as a function of the prices, p1 and p2, by q1 = 517 3.5p1 + 0.8p2, q2 = 770 4.4p2 + 1.4p1. What prices should the company charge in order to maximize the total sales revenue?

The quantity, Q, of a certain product manufactured depends on the quantity of labor, L, and of capital, K, used according to the function Q = 900L1/2K2/3. Labor costs $100 per unit and capital costs $200 per unit. What combination of labor and capital should be used to produce 36,000 units of the goods at minimum cost? What is that minimum cost?

A company manufactures a product using inputs x, y, and z according to the production function Q(x, y, z) = 20x1/2 y1/4 z2/5 . The prices per unit are $20 for x, and $10 for y, and $5 for z. What quantity of each input should be used in order to manufacture 1,200 units at minimum cost?

The Cobb-Douglas function models the quantity, q, of a commodity produced as a function of the number of workers, W, and the amount of capital invested, K: q = cW1aKa, where a and c are positive constants. Labor costs are $p1 per worker, capital costs are $p2 per unit, and there is a fixed budget of $b. Show that when W and K are at their optimal levels, the ratio of marginal productivity of labor to marginal productivity of capital equals the ratio of the cost of one unit of labor to one unit of capital.

An electrical current I flows through a circuit containing the resistors R1 and R2 in Figure 15.41. The currents i1 and i2 through the individual resistors minimize energy loss i 2 1R1 + i 2 2R2 subject to the constraint i1 + i2 = I given by Kirchoffs current law. (a) Find the currents i1 and i2 by the method of Lagrange multipliers. (b) If you are familiar with Ohms law, find the meaning of .

An open rectangular box has volume 32 cm3. What are the lengths of the edges giving the minimum surface area?

The function f(x, y) is defined for all x, y and the origin does not lie on the surface representing z = f(x, y). There is a unique point P = (a, b, c) on the surface which is closest to the origin. Explain why the position vector from the origin to P must be perpendicular to the surface at that point.

In a curious game, you and your opponent will choose three real numbers. The rules say that you must first choose a value , then your opponent is free to choose any values for x and y. Your goal is to make the value of L(x, y, ) = 10x2y22x(2x+2y) as small as you can, and your opponents goal is to make it as large as possible. What value of should you choose (assuming you have a brilliant opponent who never makes mistakes)?

An irrigation canal has a trapezoidal cross-section of area 50 m2, as in Figure 15.42. The average flow rate in the canal is inversely proportional to the wetted perimeter, p, of the canal, that is, to the perimeter of the trapezoid in Figure 15.42, excluding the top. Thus, to maximize the flow rate we must minimize p. Find the depth d, base width w, and angle that give the maximum flow rate.1

Let f(x, y) = a + x + y 1 + y + a + x , for x, y > 0, where a is a positive constant. (a) Find the critical points of f and classify them as local maxima, local minima, saddle points, or none of these. (b) Describe how the position and type of the critical points changes with respect to a, and explain this in terms of the graph of f.

Students are asked to find the global maximum of f(x, y) = x2 + y subject to the constraint g(x, y) = x2 + 2xy + y2 9=0. Student A uses the method of Lagrange multipliers with the help of a computer algebra system, and says that the global maximum is 11/4. Student B looks at a contour diagram of f and a graph of g = 0 and says there is no global maximum. Which student is correct and what mistake is the other one making?

Let f(x, y)=3x+ 2y+ 5, g(x, y)=2x2 4xy+ 5y2. (a) Find the maximum of f subject to the constraint g = 20. (b) Using the value of in part (a), estimate the maximum of f subject to each of the constraints g = 20.5 and g = 20.2. (c) Use Lagrange multipliers to find the two maxima in part (b) exactly. Compare them with the estimates.