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).Find the critical points of $f(x, y, z)=2 x^{2}+y^{2}-\frac{17}{2} z^{2}$ such that $x+y+2 z=7$ and $y-z=0$

(-5,4,4,-70)

Calculus 3

Chapter 6

An Introduction to Functions of Several Variables

Section 4

The Method of Lagrange Multipliers

Partial Derivatives

Campbell University

Oregon State University

Harvey Mudd College

University of Michigan - Ann Arbor

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

02:28

Use the method of Lagrange…

02:46

03:12

01:54

02:06

um here we wanna find critical points of two X squared plus y squared minus 17, half Z square. I'm not sure what these are. I think maybe they're like these kind of things. I'm not sure. Something like that. Hyperbole as hyperbolic. So yeah, I'm not sure what this looks like. But again we have two planes as constraints. So again these two planes will intersect in a line and that line. Um So all our solutions have to lie on that line. So that line, you know um basically passes over the surface. And basically the value of this surface along that line is what we're looking for. So we can form our Grandjean or it's not Legrand LeGrande. Gene function actually is a different thing. If you go into mechanical engineering or or physics, you'll learn about the ground function. Look Grandjean. So I don't want to call this little garage function. It's just the augmented function that has our Lagrange multipliers in it. The grand did discovered way too many things. And so his name is attached to a lot of stuff. Um So you can take Partial with respect to X, Y. Z. Landau one landed too. And those are pretty straightforward and we get five tamale co zero. So we get five equations and five unknowns and again they're linear equations. So we should have no problem solving those and we can do that back substitution. Matrix algebra, whatever. However methods you want to use. Um And what we find is X equals minus five Y equals four and Z equals four. And then here's lamp to one Atlanta to plugging those in, we get f equals -70. Now, to kind of check this, we can solve these two things. So of these two linear equations for Z. In terms of X and Y. Um no X and Y. In terms of Z. And plug those two solutions into here. And so we're now will get just a function in terms of Z. And I call that F. Bar. And then we can plot that as a function of Z. And here's what we get. And we can see our minimum occurs right here equals four, which is consistent. And then if we plug those back back into these two equations, we could find X and Y. And then our value for F. We can see is in the 70s. So again consistent with basically the two methods of of finding the optimal, either solving for the constraints and embedding them in the equation are using the Lagrange multipliers

View More Answers From This Book

Find Another Textbook

02:29

For $f(x, y)=3 x^{2}-18 x+4 y^{2}-24 y+10,$ find the point(s) at which $f_{x…

01:29

$f(x, y, z)=2 x^{2}-3 x y^{2}-2 y^{3} z^{2}+z^{2},$ determine (a) $f(1,-2,3)…

02:15

(a) Compute $\frac{d}{d x}\left(\frac{1}{b} \ln (a+b x)\right) ;$ use this t…

05:52

Determine the area of the region between the given curves.$$f(x)=2 x^{3}…

01:05

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

01:53

(a) determine a definite integral that will determine the area of the region…

01:15

Evaluate the given definite integral.(a) $\int_{a}^{b} \frac{1}{x} d x$<…

01:03

If $f^{\prime}(x)=2 e^{x},$ sketch the family of solutions.

01:13

Evaluate the given integral.$$\int x^{2} e^{4 x^{3}} d x$$

09:37

Sketch some of the level curves (contours) or the surface defined by $f$.