Problem Set 10 — Optimization

Purpose

This exercise develops your understanding of how to solve multivariable optimization problems. It therefore 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 7. Evaluate double and triple integrals.

Background

This exercise is mainly based on material in sections 3.7 and 3.8 of our textbook. We covered that material in classes between March 31 and April 7.

One exercise introduces multiple integration, drawing on sections 4.1 and 4.4 of the textbook. We covered this material in class on April 7 and 10.

Activity

Solve each of the following problems.

Problem 1

(Exercise 10 in the section 3.7E of our book.)

Find the critical points of \(f(x,y) = -x^3 + 4xy - 2y^2 + 1\) and use the second derivative test to classify them as minima, maxima, or saddle points.

Problem 2

(Exercise 2 in section 3.8E of our textbook.)

Use Lagrange multipliers to find the minimum and maximum values of \(f(x,y) = x^2y\) subject to the constraint that \(x^2 + 2y^2 = 6\).

Problem 3

(Inspired by exercise 26 in the 13.8E of our textbook.)

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

(Comment: there will be a whole circle of points around the paraboloid that all have the minimum distance to \((0,0,3)\), i.e., you will not find unique values of \(x\) and \(y\) that solve this problem.)

Problem 4

Evaluate

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, and should ordinarily last half an hour. 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.