Math 225: Calculus III    Assignment 6

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Instructions

This assignment contains problems on finding the extreme values of functions of several variables, including problems that use Lagrange multipliers, see Chapter 13 in Finney & Thomas or Chapter 3 in the Lecture Notes. Before you begin, you may want to review the notebooks MaxMin and AdvancedCalculusDemo in the Notebook folder or the back of the Lecture Notes. There are also many examples of how to do these problems in the Quizzes folder.

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Problem 1

Find the absolute maximum and minimum values of
	f[x_,y_] = x^4 -  x^2 y + x^3 + y^2
on the rectangular region bounded by the lines x == -3/2, x == 1,
y == -1/2, and y == 1.

Solution

Problem 2

Find the point(s) on the curve x y - 2x - y == 1 nearest to the point {1,2} by using Lagrange multipliers to find the minimum of the squared-distance function
f[x_,y_] = (x-1)^2 + (y-2)^2 subject to the above constraint.

Solution

Problem 3

Find the extreme values of
	f[x_,y_] = x^3 + 3 x y - y^2
on the circular disk x^2 + y^2 <= 1.

Solution

Problem 4

Suppose that the temperature T in degrees at the point {x,y,z} on the sphere
x^2 + y^2 + z^2 == 1 is T[x_,y_,z_] = 10(1-z^2) + x^2 y^2.
Locate where the highest and lowest temperatures on the sphere occur and what the temperatures are at those points. (Use Lagrange multipliers and be sure to ignore any complex numbers that arise.)

Solution

Problem 5

Your firm has been asked to design a storage tank for liquid petroleum gas. The customer's specifications call for a cylindrical tank that holds 8000 m^3 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 tank?

Solution