Exam 18: Multivariable Calculus

Determine the critical points of f(x,y)= 3 + 4 - 2x + 8y and also determine by the second-derivative test whether each point corresponds to a relative maximum,to a relative minimum,to neither,or whether the test gives no information.

The Cobb-Douglas production function for a company is given by P(k,l)= 65 where P is the monthly production value when k is the number of units of capital and l is the number of units of labor.Suppose that capital costs $60 per unit,labor costs $140 per unit,and the total cost of capital and labor is limited to $70,000.Use Lagrange multipliers to write the system of equations you would use to find the number of units of capital and labor that maximize production.

An empirical formula relating the surface area A (in square inches)of an average human body to the weight w (in pounds)and the height h (in inches)of the person is A(w,h)= 15.64 .Find Then find and interpret (155,66)

Evaluate:

An empirical formula relating the surface area A (in square inches)of an average human body to the weight w (in pounds)and the height h (in inches)of the person is A(w,h)= 15.64 . Find and .

Use the method of Lagrange multipliers to determine the critical points of f(x,y)= x + 2y subject to the constraint xy = 8.

The Cobb-Douglas production function for a company is given by P = 20 ,where P is the monthly production value when k is the amount of the company's capital investment (in dollars per month)and l is the size of the labor force (in work hours per month).Find and .

To fill an order for 100 units of a product,a firm wishes to distribute the production between its two plants,Plant 1 and Plant 2.The total cost function is given by c = f( , )= 0.5 + 2 + 32 + 500,where and are the number of units produced at Plants 1 and 2,respectively.How should the output be distributed in order to minimize costs?

If f(x,y)= ,find: (a) (x,y) (b) (x,y) (c) (x,y)

An empirical formula relating the surface area A (in square inches)of an average human body to the weight w (in pounds)and the height h (in inches)of the person is A(w,h)= 15.64 .Find Then find and interpret (105,64).

An empirical formula relating the surface area A (in square inches)of an average human body to the weight (in pounds)and the height h (in inches)of the person is A(w,h)= 15.64 .Find and .

If f(x,y)= find

Use the method of Lagrange multipliers to determine the critical points of f(x,y,z)= + 4y - subject to the constraint x + 2y - 4z = 3.

The Cobb-Douglas production function for a company is given by where P is the monthly production value when k is the number of units of capital and l is the number of units of labor.Suppose that capital costs $150 per unit,labor costs $225 per unit,and the total cost of capital and labor is limited to $270,000.Use Lagrange multipliers to write the system of equations you would use to find the number of units of capital and labor that maximize production.

An open rectangular cardboard box is to have a volume of 4 cubic feet.Find the dimensions of the box so that the amount of cardboard is minimized.

If f(x,y,z)= ,find (a) (x,y,z),(b) (x,y,z),and (c) (x,y,z).

Determine all of the critical points of f(x,y)= + 3 - 9x + - 12y.Also use the second derivative test to determine,if possible,whether a maximum,minimum or saddle point occurs at each of these critical points.

Evaluate:

Find and where f(x,y)= .

Evaluate: