Problem Set 9 — Optimization

Purpose

This exercise concentrates on finding optimal values for multivariable functions subject to constraints. In doing so, it also reinforces your ability to find partial derivatives and work with Mathematica. It thus contributes to the following learning outcomes for this course:

• Outcome 5. Compute derivatives of functions of 2 and 3 variables
• Outcome 6.2. Apply derivative concepts to solve optimization problems
• Outcome 12. Use technological tools such as computer algebra systems or graphing calculators for visualization and calculation of multivariable calculus concepts.

Background

This exercise is based on material in sections 3.7 and 3.8 in our textbook. We covered those sections in classes between October 21 and October 28.

The problem set also asks you to use Mathematica to help solve some of its problems. Two recently discussed features of Mathematica will be particularly helpful, namely the ability to define your own functions, and the built-in Solve function for solving equations. We will discuss these features in class on October 31.

Activity

Solve each of the following problems.

Problem 1

Consider the production function

with the constraint

In addition to this constraint, assume that \(x\) and \(y\) must always be greater than or equal to 0.

Part A

Use critical points and the values of \(f\) along the boundaries of the region in which \(x\) and \(y\) must lie to find the values of \(x\) and \(y\) that maximize production. Find the value of \(f\) at that point. You may use Mathematica to help you with the calculations and record-keeping in this problem.

Part B

Use Lagrange multipliers to find the values of \(x\) and \(y\) that maximize production. Confirm that you find the same values as you found in Part A.

Problem 2

(Exercise 26 in the 13.8E of our textbook.)

Find the minimum distance from the parabola \(y = x^2\) to point \((0,3)\).

Problem 3

Consider the four-variable function

Part A

Find the critical point (there’s only one) for \(f\).

Part B

I don’t know that there’s a second derivative test to decide what happens at a critical point of a 4-variable function, but try to think of, and informally justify, an analog of the first-derivative test from single-variable calculus that you could use to justify believing that the critical point you found in Part A corresponds to a local minimum.

Follow-Up

I will grade this exercise during an individual meeting with you. That meeting should happen on or before the “Grade By” date above. During the meeting I will look at your solution, ask you any questions I have about it, answer questions you have, etc. Sign up for the meeting via Google calendar. Please have a written solution to the exercise ready to share with me during your meeting, as that will speed the process along.