Exam 8: Functions of Several Variables

Find all the critical points of the function Classify these critical points as local maxima, local minima, or neither.

A company manufactures x units of one item and y units of another. The total cost, C, in dollars of producing these two items is given by the function . Use Lagrange multipliers to find the minimum cost subject to the constraint that 100 items (total) must be produced. What is the value of to one decimal place?

You build a campfire while up in the mountains. It is 45 F when you start the fire. Let be the temperature x feet from the fire t minutes after you start it. The following figure shows the contour diagram for H. Is H an increasing or decreasing function of x?

A mop company can produce mops using x units of capital and y units of labor, with production costs dollars. With a budget of $400,000, the maximum production is 65,000, using $300,000 in capital and $100,000 in labor. The Lagrange multiplier is . What are the units for ?

The following is a contour diagram for . Estimate f_x .

If , .

The monthly mortgage payment in dollars, P, for a house is a function of three variables P = f(A, r, N), where A is the amount borrowed in dollars, r is the interest rate, and N is the number of years before the mortgage is paid off. It is given that: Estimate the value of and interpret your answer in terms of a mortgage payment. Select all answers that apply.

Which of the following describes the contour diagram for the function (with m on the vertical axis and d on the horizontal axis)?

A company sells two products. The fixed costs for the company are $2500, the variable costs for the first product are $5 per unit, and the variable costs for the second product are $9 per unit.. If the company produces units of the first product and units of the second product, and the total cost is given by , what is ?

Estimate f_y (3,2) from the contour diagram shown below. Use the next higher point to make your estimate.

The amount, , of your paycheck is a function of your hourly wage, w, and the number of hours, h, you work. What does the statement mean in terms of your paycheck?

Use Lagrange multipliers to find the maximum and minimum values of , subject to the constraint . For the minimum value, = _____ (use the decimal form), and the minimum value of is _____.

The function has a local __________ (maximum / minimum / neither) at the critical point, where x = _____ and y = _____.

The monthly car payment for a new car is a function of three variables, , where r is the annual interest rate, A is the amount borrowed in dollars, and t is time in months before the car is paid off. Would you expect f_A to be increasing or decreasing?

The function where a and b are constants is sometimes referred to as a "bump function" and is used to construct functions which take on maximum values at certain points. Show that f(x, y) has a maximum at (a, b).

Find if .

Find the maximum value of subject to the constraint equation using Lagrange multipliers.

Suppose that you want to find the maximum and minimum values of subject to the constraint x + 6y = 4. Use the method of Lagrange multipliers to find the exact location(s) of any extrema.

Yummy Potato Chip Company has manufacturing plants in N.Y. and N.J. The cost of manufacturing depends on the quantities (in thousand of bags), q₁ and q₂, produced in the N.Y. and N.J. factories respectively. Suppose the cost function is given by Find .

The quantity, Q, of a good produced depends on the quantities and of the two main materials used: . Material costs $75 per unit, and material costs $25 per unit. We want to minimize the cost of producing 100 units of the good. What is the constraint function?