Download the App!

Get 24/7 study help with the Numerade app for iOS and Android! Enter your email for an invite.

Get the answer to your homework problem.

Try Numerade free for 7 days

Like

Report

Use the method of Lagrange multipliers to optimize $f$ as indicated, subject to the given constraint(s).Maximize $f(x, y, z)=x y z$ such that $x y+2 x z+2 y z=8$

$\frac{8 \sqrt{6}}{9}$

Calculus 3

Chapter 6

An Introduction to Functions of Several Variables

Section 4

The Method of Lagrange Multipliers

Partial Derivatives

Johns Hopkins University

Missouri State University

University of Nottingham

Boston College

Lectures

12:15

In calculus, partial derivatives are derivatives of a function with respect to one or more of its arguments, where the other arguments are treated as constants. Partial derivatives contrast with total derivatives, which are derivatives of the total function with respect to all of its arguments.

04:14

01:49

Use the method of Lagrange…

02:38

02:06

01:30

03:12

01:59

03:53

02:14

Yeah. Mhm. And this taste Well you can't want to uh see your best to maximize or minimize or what do we want to maximize six times Y times Z. Subject to this constraint, which I don't know what kind of surface that is, but we could obviously plot it if we wanted to but its X times Y plus two X times E plus two Y times Z minus +80. That's our constraint equation. The constraint function is just this. So G is X times Y M C plus lambda times are constraint function. Yeah. And we can take derivatives with respect to X. Y. Z. And lambda. And these are just paranormal. Also they aren't too bad. And so we get with respect to X is Y times Z plus land at times Y plus two Z. And that equals zero. Um Director, why we get Xz plus Lander times X plus two Z. And we set that to zero. Have set to see, we get X, Y plus two lambda times X plus Y. And we set that equal zero respect to land up. And we just get our constraint equation back when we set it equal to zero. So again, we have four equations of foreign loans, they're not linear. So it means that there's possibly multiple solutions here. And we have um you know, these are always going to be linear and lambda. So it's always easy. It's best always kind of try to eliminate land at first and then to keep eliminating whatever is the easiest thing to eliminate after that. So um you know, after a bunch of back substitution in algebra we wind up with two solutions. Um Either uh X is minus two times the square to two thirds. And why is the same value both solutions, X equals Y. But they're either this plus or minus or plus or minus two times the square root of two thirds. Okay, so we can take both of these and plug them in. Uh huh. Uh Wait a minute I z we have Z. Um they all have the same sign, right? So this is one solution and this is the other solution, you can take this and plug it into here and then obviously there's and plug it in and we get f equals has two different values minus eight thirds times the square root of two thirds plus eight thirds times square to two thirds. And we wanted to find a maximum. So this is of course our maximum right here. Um This point here X equals two times the square the two thirds Y equals two times the square two thirds, and Z equals the square two thirds

View More Answers From This Book

Find Another Textbook

01:05

$f(x, y, z)=100 x^{1 / 2} y^{1 / 3} z^{1 / 6},$ determine (a) $f(4,8,1),(\ma…

02:44

Find the partial derivatives with respect to (a) $x$ and (b) $y$.$$f(x, …

05:03

Evaluate the given integral.$$\int \frac{(\ln x)^{3}}{x} d x$$

02:22

Evaluate the given integral and check your answer.$$\int \frac{4 x^{3}-7…

01:32

Evaluate the given integral.$$\int \frac{e^{2 x}}{e^{2 x}+1} d x$$

03:06

Find and classify, using the second partial derivative test, the critical po…

01:52

Evaluate the given integral.$$\int \sqrt{2 x+1} d x$$

Use the method of Lagrange multipliers to optimize $f$ as indicated, subject…

02:55

Evaluate the given definite integral.$$\int_{0}^{1}(3 x-2)^{4} d x$$