Using the Lagrange multiplier to find extrema?

[CalleighMay]

Hey guys! I have been on the forum for about a week or so and have compiled a lot of information and techniques to help me understand calculus, so i really appreciate everyone's help!

 

I am a soon-to-be freshman in college and am taking a summer class, calculus II (took calc I in HS). This is our last week of class after our final exam so my professor is taking this time to give us a preview of what we will be learning in the fall semester in Calc III (since this is the same professor). Every Tuesday class our professor gives us a few problems from future sections and asks us to "see what we can come up with" and to work together to find solutions. The following Tuesday he asks us to discuss the problems as a class, seeing which ones of us know our stuff =P

 

Basically, i want to ask you guys what you think about these problems as i do them along before i have my discussion. I really want to make a lasting impression on my professor by "knowing my stuff" -to show him i can do it! All's i need is a little help! Would you guys mind giving me some help?

 

We are using the textbook Calculus 8th edition by Larson, Hostetler and Edwards and the problems come from the book.

 

The problem is on pg 974 in chapter 13.10 in the text, number 10. It reads:

 

Use Lagrange multipliers to find the indicated extrema, assuming that x and y are positive.

And it gives:

minimize: f(x,y)= sqrt(x^2+y^2) with the constraint: 2x+4y-15=0

 

I looked at similar problems in the same section but they're little help. I did the best i could so far and this is what i came up with.

 

subject to the constraint:

2x+4y=15

let g(x,y)=2x+4y=15

 

since tri f(x,y)=x/sqrt(x^2+y^2)

I got this step, then in the next step i'm supposed to use some symbol that looks like, well i remember it as wavelength in physics, kinda looks like a k??

and tri g(x,y)=2(wavelength symbol)i+4(wavelength symbol)j

 

but to attain a system of linear equations, how do i do that with x/sqrt(x^2+y^2)??

 

Any help would be greatly appreciated. Thanks guys

[Dr. Jekyll]

First note that you can chose to minimize [math](x^2+y^2)[/math] instead of [math]\sqrt(x^2+y^2)[/math] (they have the same minima).

 

Doing Lagrange relaxation:

 

[math]

\mathcal{L}(x,y,\lambda)=x^2+y^2+\lambda(2x+4y-15)

[/math]

 

For extreme value we have [math]\nabla \mathcal{L}=0[/math] and

 

[math]\nabla \mathcal{L}=[2x+2\lambda, 2y+4\lambda, 2x+4y-15]=[0, 0, 0][/math].

 

Three equations and three variables to solve for ( then you might want to check if it is a minimum of maximum).

[Pete]

Hi CalleighMay

 

Welcome to the forum. I'm glad to see young adults such as yourself work so hard at things like this. Its refreshing!

 

One of my majors as an undergraduate was mathematics. I was also a tutor in my college's math lab for several years. Please feel to PM me anytime with any question that you have. I'd be glad to help in anyway I can. That applies to everyone on this forum too.

 

I like to see students work so hard as you are and trying to impress their teachers. That's what I did in college.

 

Good luck

 

Pete

Edited August 26, 2008 by Pete
multiple post merged