Lagrange multipliers Definition and 173 Threads

Hey all, this is my first post, so I apologize in advance if data are missing/format is strange/etc. I'm working with lagrange multipliers, and I can get to the answer about half the time. The problem is, I'm not really sure how to deal with things when the multiplier equation becomes...

Homework Statement Find the minimum of f(x,y) = x^2 + y^2 subject to the constraint g(x,y) = xy-3 = 0 Homework Equations delF = lambda * delG The Attempt at a Solution Okay, after lecture, reviewing the chapter and looking at some online information, this is what I have so far...

Hi There I would like help on a question about Lagrange multipliers. Question: Consider the intersection of two surfaces: an elliptic paraboloid z=x^2 + 2*x + 4*y^2 and a right circular cylinder x^2 + y^2 = 1. Use Lagrange multipliers to find the highest and lowest points on the curve of the...

Hello physicsforums community. I have recently learned about Lagrange multipliers and have been given three problems to solve. Could you guys please go over my work and see if I have the gist of it? One question, a theoretical one, I have no idea how to begin. Any advice regarding this would be...

Find the maximum and minimum of f(x,y)=y2-x2 with the constraint x2/4 +y2=2. My calculus professor gave us this on his exam and there were no problems like this in the book and I would just like to know how it's done because it's bothering me ha. After doing the partial derivatives I got...

Homework Statement Find 3 positive numbers x, y and z for which: their sum is 24 and which maximizes the product: P = x2y3z. Find the maximum product. The Attempt at a Solution Ok, I know how to set up the equations. x + y + z = 24 Delta(F) <2xy3z, 2x2y2z, x2y3> fx = 2xy3z...

Homework Statement Minimise = x2 + y2 subject to C(x,y) = 4x2 + 3y2 = 12. Homework Equations The Attempt at a Solution I let h(x,y) = x2 + y2 + \lambda(4x2 + 3y2 - 12). I got hx = 2x + 8\lambdax = 0, hy = 2y + 6\lambday = 0, but here I get 2 values of \lambda, \lambda = -1/4 &...

I do not have one specific question that needs answering. Rather, it is about Lagrange multipliers in general. So for certain minimization/maximization questions (ie find the shortest distance from some point to some plane) it seems that one could solve the question using lagrange multipliers...

Lagrange Multipliers Question? Homework Statement Find the minimum and maximum values of the function subject to the given constraint. f (x,y,z) = x^2 - y - z, x^2 - y^2 +z = 0 The Attempt at a Solution Okay this is what I did: Gradient f = <2x,-1,-1> Gradient g =...

Homework Statement See figure Homework Equations N/A The Attempt at a Solution Alright we'll this is my first shot at a question like this, so in all honesty I don't know what concepts this question is testing. It mentions finding absolute max/min of a function inside a...

Hi, I'm trying to do a constrained optimization problem. I shall omit the details as I don't think they're important to my issue. Let f:\mathbb R^n \to \mathbb R and c:\mathbb R^n \to \mathbb R^+\cup\{0\} be differentiable functions, where \mathbb R^+ = \left\{ x \in \mathbb R : x> 0...

Homework Statement Using Lagrange multipliers, find the maximum and minimum values of f(x,y)=x^3y with the constraint 3x^4+y^4=1.Homework Equations The Attempt at a Solution Here is my complete solution. I just wanted to make sure there are no errors and I did it correctly. Thanks for any...

I have been reading about Lagrange Multipliers, my book along with wiki and other resources I have read use an intuitive argument on why the max/min contour lines end up tangent to the constraint equation. I don't really understand it, especially considering the obvious flaw as shown by the...

Homework Statement Find the points on the ellipse x2 + 2y2 = 1 where f(x,y) = xy has its extreme values. Homework Equations The Attempt at a Solution f(x,y,z) = x2 + y2 + z2 -- constraint g(x,y,z) = x2 + 2y2 -1 = 0 gradient of f = \lambda * gradient of g 2xi + 2yj + 2zk =...

Homework Statement A point lies on the plane x-y+z=0 and on the ellipsoid x^2 +\frac{y^2}{4} + \frac{z^2}{4} = 1 Find the minimum and maximum distances from the origin of this point. The Attempt at a Solution The two contraints g = x-y+z =h= x^2 +\frac{y^2}{4} +...

Homework Statement find the points on the surface x^2-z^2 = 1 which are in minimum distance from (0,0) i should find the points using d = x^2+y^2+z^2 first of all gradf = λ gradg where f = d and g = x^2-z^2 so we have (2x,2y,2z) = λ (2x,0,2z) now 2x = λ2x 2y = 0 => y = 0 2z = λ2z so...

Homework Statement Find the extrema of the given function subject to the given constraint: f(x,y)=x2-2xy+2y2, subject to x2+y2=1Homework Equations Lagrange Multipliers The Attempt at a Solution First, I defined the constraint to be g(x,y)=0, that is, g(x,y)=x2+y2-1 I then set up the usual...

Hey folks. :smile: I have some more or less qualitative questions regarding optimization problems via Lagrange multipliers. I am following the http://en.wikipedia.org/wiki/Lagrange_multipliers" on this one and I am just a little confused by their wording. In the first section titled...

Homework Statement Assume that the surface temperature distribution of an ellipsoid shaped object given by 4x2 + y2 + 4z2 = 16 is T(x,y,z) = 8x2 + 4yz - 16z + 600.Homework Equations The Attempt at a Solution I'm assuming we just have to find the maximum value of this function using the lagrange...

Homework Statement Why is \nabla f = \lambda \nabla g where f is the function you want to find the extrema of and g is the contraint? Also how would you identify the above in the following Determine the least real number M such that the inequality |ab(a^2-b^2) +...

determine, if any, the maximum and minimum values of the scalar field f (x, y) = xy subject to the constraint 4x^2{}+9y^2{}=36 The attempt at a solution using Lagrange multipliers, we solve the equations \nablaf=\lambda\nablag ,which can be written as f_{x}=\lambdag_{x}...

Homework Statement f(x,y,z)=exy and x5+y5=64 Find Max and MinHomework Equations ∇F = <yexy, xexy> λ∇G = <5x4λ, 5y4λ> The Attempt at a Solution yexy = 5x4λ xexy = 5y4λ x5+y5=64 No idea where to go from here...

Homework Statement Use lagrange multipliers to find the shortest distance between a point on the elliptic paraboloid z=x^2 +y^2 Homework Equations The Attempt at a Solution http://img716.imageshack.us/img716/7272/cci1902201000000.jpg I'm not that good with using the equation...

Homework Statement Find the point on 2x + 3y + z - 11 = 0 for which 4x^2 +y^2 +z^2 is a minimum Homework Equations The Attempt at a Solution Using lagrange multipliers I find: F = 4x^2 + y^2 + z^2 + l(2x + 3y + z) Finding the partial derivatives I get the three equations...

Homework Statement By using the Lagrange multipliers find the extrema of the following function: f(x,y)=x+y subject to the constraints: x2+y2+z2=1 y+z=12. The attempt at a solution Using lambda = 1/(2x) I got x=y-z and y=1-z plugging that into the first constraint, I got: 6y^2-6y+1=0 which...

Homework Statement Use Lagrange Multipliers to find the points closest to the origin on the curve defined implicitly by x2-xy+y2-z2 = 1 x2+y2=1 2. The attempt at a solution I know how to do this for regular curves, but I don't know where to start with implicitly defined ones. Any...

Homework Statement Using Lagrange Multipliers, we are to find the maximum and minimum values of f(x,y) subject to the given constraint Homework Equations f(x,y,z) = x^2 - 2y + 2z^2, constraint: x^2 + y^2 + z^2 = 1 The Attempt at a Solution grad f = lambda*grad g (2x, -2, 4z) =...

Homework Statement Find the absolute maximum and minimum values for f(x,y) = sin x + cos y on the rectangle R defined by 0<=x<=2pi and 0<=y<=2pi using the method of Lagrange Multipliers. The Attempt at a Solution I don't know where to start in getting the constraint into something I...

Let F and f be functions of the same n variables where F describes a mechanical system and f defines a constraint. When considering the variation of these functions why does eliminating the nth term (for example using the Lagrange multiplier method) result in a free variation problem where it...

Homework Statement Consider the problem of finding the points on the surface xy+yz+zx=3 that are closest to the origin. 1) Use the identity (x+y+z)^2=x^2+y^2+z^2+2(xy+yz+zx) to prove that x+y+z is not equal to 0 for any point on the given surface. 2) Use the method of Lagrange...

Homework Statement Use Lagrange Multipliers to find the Maximum and Minimum values of f(x,y) = x2-y. Subject to the restraint g(x,y) = x2+y2=25 Homework Equations gradient f(x,y)= gradient g(x,y) The Attempt at a Solution I have found the gradients of f and g to be f(x,y) =...

1. Homework Statement [/b] f\left(x,y\right) = x^2 +y^2 g\left(x,y\right) = x^4+y^4 = 2 Find the maximum and minimum using Lagrange multiplier Homework Equations The Attempt at a Solution grad f = 2xi +2yj grad g= 4x^3i + 4y^3j grad f= λ grad g 2x=4x^3λ and 2y=...

help me out on this proble i am confuse a sport center is to be constructed.it consists of a rectangular region with a semicircle ach end .if the perimater of the room is to be a 500 meter running truck find the dimetion that will make the area as large as possible. i can find if the...

Homework Statement A disk moves on an inclined plane, with the constraint that it's velocity is always at the same direction as it's plane (similar to an ice skate, maybe). In other words: If \hat{n} is a vector normal to the disk's plane, we have at all times: \hat{n} \cdot \vec{v} = 0. Also...

Homework Statement Find the maximum and minimum values of f = (x-1)^2 + (y-1)^2 on the boundary of the circle g = x^2 + y^2 = 45. Homework Equations f=(x-1)^2 + (y-1)^2 g=x^2+y^2=45 gradf(x,y)=lambda*gradg(x,y) The Attempt at a Solution gradf(x,y)=<2x-2,2y-4>...

Homework Statement Point P(x,y,z) lies on the part of the ellipsoid 2x^2 + 10y^2 + 5z^2 = 80 that is in the first octant of space. It is also a vertex of a rectangular parallelpiped each of whose sides are parallel to a coordinate plane. Use Method of LaGrange Multipliers to determine the...

Homework Statement Use lagrange multipliers to find the maximum and minimum values of f subject to the given constraint, if such values exist. f(x,y) = x+3y, x2+y2≤2 Homework Equations grad f = λ grad g The Attempt at a Solution to find critical points in the interior region...

Homework Statement A particle of mass, m1, is constrained to move in a circle with radius a at z=0 and another particle of mass, m2, moves in a circle of radius b at z=c. For this we wish to write up the Lagrangian introucing the constraints by lagrange multipliers and solve the following...

I'm trying to follow the idea behind Lagrange multipliers as given in the following wikipedia link. http://en.wikipedia.org/wiki/Lagrange_multipliers I follow the article right up until the point where it goes: 'To incorporate these conditions into one equation, we introduce an...

Homework Statement Using the method of lagrange multipliers, find the points on the curve 3x² - 4xy + 6y² = 140 which are closest and furthermost from the ORIGIN and the corresponding distances between them The Attempt at a Solution I have done roughly half the question but appear to be...

find min/max: f(x,y)=xy with constraint being 4x^2+9y^2=32 [gradient]f=[lambda]gradient g The Attempt at a Solution I thought I understood the Lagrange problems, but I can't seem to get the minimum right on the last few problems. I get x=+/-2 and then plug back into find y, then I use my...

I have used this method quite a lot but I have never completely understood the proof. The only book I have that provides a proof is Shifrin's "Multivariable Mathematics" which I find kind of confusing. Stewart's "proof" is more or less just geometric intuition. Does anyone know of a book that...

Hello: Problem1: The temp of the circular plate D= {(x1,x2) | x1^{2} + x2^{2} \leq 1} is given by T=2x^{2} -3y^{2} - 2x. Find hottest and coldest points of the plate. Problem 2 Show that for all (x1,x2,x3) \in R^{3} with x1>0, x2>0, x3>0 and x1x2x3 = 1, we have x1+x2+x3 \geq3...

We're suppose to minimize f(x,y,z)=x^2+y^2+z^2 subject to 2x+y+2z=9. I only ever remember learning how to do f(x,y) would it be the same equation? Thus, f(x,y,\lambda) = f(x,y) + \lambda g(x,y)? Meaning f(x,y,z,\lambda) = x^2+y^2+z^2 + \lambda (2x+y+2z-9) and then continue solving for each...

http://www.geocities.com/asdfasdf23135/advcal29.JPG I am wondering whether the above statement is true. "A necessary condition for the constrained optimization problem to have a GLOBAL min or max is that..." Should the word local replace global? I am confused about the method of...

(1) f(x,y,z)=x+2y (2) x+y+z=1 (3) y^2+z^2=4 1=\lambda 2=\lambda+2y\mu 0=\lambda+2z\mu u=\frac{1}{2y} y=\pm\sqrt2 \ \ \ z=\pm\sqrt2 Plugging into equation 2 to solve for x. How do I know to use either y=\sqrt 2 \ \mbox{or} \ y=-\sqrt2 ... similarly with my values for z. edit: NVM, I'm an...

Homework Statement The Park Service is building shelters for hikers along the Appalachian Trail. Each shelter has a back, a top, and two sides. Find the dimensions that will maximize the volume while using 384 square feet of wood. They want me to find the length, width, and height...

I'm having a little trouble with another old test question. It states: Use LaGrange multipliers to find the point on the line 2x + 3y = 3 that is closest to the point P(4, 2). I assume that my constraint is g(x, y) = 2x + 3y = 3, and I have to come up with a function f(x, y) to be...

Homework Statement A uniform hoop of mass m and radius r rolls without slipping on a fixed cylinder of radius R. The only external force is that of gravity. If the smaller cylinder starts rolling from rest on top of the bigger cylinder , use the method of lagrange multipliers to find the point...

Hi! I've been studying Dirac's programme for some time and I realized that there's something missing: Actually this is missing in every standard book on classical mechanics concerning how constraints are implemented in the lagrangian. They are usually inserted with some unknown variables...